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#### Question 1:

Prove that the function f(x) = loge x is increasing on (0, ∞).

#### Question 2:

Prove that the function f(x) = loga x is increasing on (0, ∞) if a > 1 and decreasing on (0, ∞), if 0 < a < 1.

#### Question 3:

Prove that f(x) = ax + b, where a, b are constants and a > 0 is an increasing function on R.

#### Question 4:

Prove that f(x) = ax + b, where a, b are constants and a < 0 is a decreasing function on R.

#### Question 5:

Show that f(x) = $\frac{1}{x}$ is a decreasing function on (0, ∞).

#### Question 6:

Show that f(x) = $\frac{1}{1+{x}^{2}}$ decreases in the interval [0, ∞) and increases in the interval (−∞, 0].

#### Question 7:

Show that f(x) = $\frac{1}{1+{x}^{2}}$ is neither increasing nor decreasing on R.

#### Question 8:

Without using the derivative, show that the function f (x) = | x | is
(a) strictly increasing in (0, ∞)
(b) strictly decreasing in (−∞, 0).

#### Question 9:

Without using the derivative show that the function f (x) = 7x − 3 is strictly increasing function on R.

#### Question 1:

Find the intervals in which the following functions are increasing or decreasing.
(i) f(x) = 10 − 6x − 2x2

(ii) f(x) = x2 + 2x − 5

(iii) f(x) = 6 − 9x − x2

(iv) f(x) = 2x3 − 12x2 + 18x + 15

(v) f(x) = 5 + 36x + 3x2 − 2x3

(vi) f(x) = 8 + 36x + 3x2 − 2x3

(vii) f(x) = 5x3 − 15x2 − 120x + 3

(viii) f(x) = x3 − 6x2 − 36x + 2

(ix) f(x) = 2x3 − 15x2 + 36x + 1

(x) f(x) = 2x3 + 9x2 + 12x + 20

(xi) f(x) = 2x3 − 9x2 + 12x − 5

(xii) f(x) = 6 + 12x + 3x2 − 2x3

(xiii) f(x) = 2x3 − 24x + 107

(xiv) f(x) = −2x3 − 9x2 − 12x + 1

(xv) f(x) = (x − 1) (x − 2)2

(xvi) f(x) = x3 − 12x2 + 36x + 17

(xvii) f(x) = 2x3 − 24x + 7

(xviii) $f\left(x\right)=\frac{3}{10}{x}^{4}-\frac{4}{5}{x}^{3}-3{x}^{2}+\frac{36}{5}x+11$

(xix) f(x) = x4 − 4x

(xx) $f\left(x\right)=\frac{{x}^{4}}{4}+\frac{2}{3}{x}^{3}-\frac{5}{2}{x}^{2}-6x+7$

(xxi) f(x) = x4 − 4x3 + 4x2 + 15

(xxii) f(x) = $5{x}^{\frac{3}{2}}-3{x}^{\frac{5}{2}}$x > 0

(xxiii) f(x) = x8 + 6x2

(xxiv) f(x) = x3 − 6x2 + 9x + 15

(xxv) $f\left(x\right)={\left\{x\left(x-2\right)\right\}}^{2}$

(xxvi) $f\left(x\right)=3{x}^{4}-4{x}^{3}-12{x}^{2}+5$

(xxvii) $f\left(x\right)=\frac{3}{2}{x}^{4}-4{x}^{3}-45{x}^{2}+51$

(xxviii)

(xxix) $f\left(x\right)=\frac{{x}^{4}}{4}-{x}^{3}-5{x}^{2}+24x+12$

#### Answer:                                                  (xxix)

Thus, for the increasing function the interval is $\left(-3,2\right)\cup \left(4,\infty \right)$ and for the decreasing function $\left(-\infty ,-3\right)\cup \left(2,4\right)$.

#### Question 2:

Determine the values of x for which the function f(x) = x2 − 6x + 9 is increasing or decreasing. Also, find the coordinates of the point on the curve y = x2 − 6x + 9 where the normal is parallel to the line y = x + 5.

#### Answer:

Let (x, y) be the coordinates on the given curve where the normal to the curve is parallel to the given line.
Slope of the given line = 1

#### Question 3:

Find the intervals in which f(x) = sin x − cos x, where 0 < x < 2π is increasing or decreasing.

#### Question 4:

Show that f(x) = e2x is increasing on R.

#### Question 5:

Show that f(x) = e1/x, x ≠ 0 is a decreasing function for all x ≠ 0.

#### Question 6:

Show that f(x) = loga x, 0 < a < 1 is a decreasing function for all x > 0.

#### Question 7:

Show that f(x) = sin x is increasing on (0, π/2) and decreasing on (π/2, π) and neither increasing nor decreasing in (0, π).

#### Question 8:

Show that f(x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π).

#### Question 9:

Show that f(x) = x − sin x is increasing for all xR.

#### Question 10:

Show that f(x) = x3 − 15x2 + 75x − 50 is an increasing function for all xR.

#### Question 11:

Show that f(x) = cos2 x is a decreasing function on (0, π/2).

#### Question 12:

Show that f(x) = sin x is an increasing function on (−π/2, π/2).

#### Question 13:

Show that f(x) = cos x is a decreasing function on (0, π), increasing in (−π, 0) and neither increasing nor decreasing in (−π, π).

#### Question 14:

Show that f(x) = tan x is an increasing function on (−π/2, π/2).

#### Question 15:

Show that f(x) = tan−1 (sin x + cos x) is a decreasing function on the interval (π/4, π/2).

#### Question 16:

Show that the function f(x) = sin (2x + π/4) is decreasing on (3π/8, 5π/8).

#### Question 17:

Show that the function f(x) = cot$-$l(sinx + cosx) is decreasing on $\left(0,\frac{\mathrm{\pi }}{4}\right)$ and increasing on $\left(\frac{\mathrm{\pi }}{4},\frac{\mathrm{\pi }}{2}\right)$.

#### Question 18:

Show that f(x) = (x − 1) ex + 1 is an increasing function for all x > 0.

#### Question 19:

Show that the function x2x + 1 is neither increasing nor decreasing on (0, 1).

#### Question 20:

Show that f(x) = x9 + 4x7 + 11 is an increasing function for all xR.

#### Question 21:

Prove that the function f(x) = x3 − 6x2 + 12x − 18 is increasing on R.

#### Question 22:

State when a function f(x) is said to be increasing on an interval [a, b]. Test whether the function f(x) = x2 − 6x + 3 is increasing on the interval [4, 6].

#### Question 23:

Show that f(x) = sin x − cos x is an increasing function on (−π/4, π/4).

#### Question 24:

Show that f(x) = tan−1 xx is a decreasing function on R.

#### Question 25:

Determine whether f(x) = −x/2 + sin x is increasing or decreasing on (−π/3, π/3).

#### Question 26:

Find the intervals in which f(x) = log (1 + x) − $\frac{x}{1+x}$ is increasing or decreasing.

#### Question 27:

Find the intervals in which f(x) = (x + 2) ex is increasing or decreasing.

#### Question 28:

Show that the function f given by f(x) = 10x is increasing for all x.

#### Question 29:

Prove that the function f given by f(x) = x − [x] is increasing in (0, 1).

#### Question 30:

Prove that the following functions are increasing on R.
(i) f$\left(x\right)=$3${x}^{5}$ + 40${x}^{3}$ + 240$x$
(ii) $f\left(x\right)=4{x}^{3}-18{x}^{2}+27x-27$

#### Answer:

(i)

So, f(x) is increasing on R.

(ii) $f\left(x\right)=4{x}^{3}-18{x}^{2}+27x-27$

So, f(x) is increasing on R.

#### Question 31:

Prove that the function f given by f(x) = log cos x is strictly increasing on (−π/2, 0) and strictly decreasing on (0, π/2).

#### Question 32:

Prove that the function f given by f(x) = x3 − 3x2 + 4x is strictly increasing on R.

#### Question 33:

Prove that the function f(x) = cos x is:
(i) strictly decreasing in (0, π)
(ii) strictly increasing in (π, 2π)
(iii) neither increasing nor decreasing in (0, 2π)

#### Question 34:

Show that f(x) = x2x sin x is an increasing function on (0, π/2).

#### Question 35:

Find the value(s) of a for which f(x) = x3ax is an increasing function on R.

#### Question 36:

Find the values of b for which the function f(x) = sin xbx + c is a decreasing function on R.

#### Question 37:

Show that f(x) = x + cos xa is an increasing function on R for all values of a.

#### Question 38:

Let f defined on [0, 1] be twice differentiable such that | f"(x) | ≤ 1 for all x ∈ [0, 1]. If f(0) = f(1), then show that | f'(x) | < 1 for all x ∈ [ 0, 1].

#### Answer:

If a function is continuous and differentiable and f(0) = f(1) in given domain x ∈ [0, 1],
then by Rolle's Theorem;
f'(x) = 0 for some x ∈ [0, 1]
Given: |f"(x)| ≤ 1
On integrating both sides we get,
|f'(x)| ≤ x
Now, within interval x ∈ [0, 1]
We get, |f' (x)| < 1.

#### Question 39:

Find the intervals in which f(x) is increasing or decreasing:

(i) f(x) = x|x|, x $\in$R

(ii) f(x) = sinx + |sinx|, 0 < x $\le 2\mathrm{\pi }$

(iii) f(x) = sinx(1 + cosx), 0 < x < $\frac{\mathrm{\pi }}{2}$
[CBSE 2014]

#### Question 1:

The interval of increase of the function f(x) = xex + tan (2π/7) is
(a) (0, ∞)
(b) (−∞, 0)
(c) (1, ∞)
(d) (−∞, 1)

(b) (−∞, 0)

#### Question 2:

The function f(x) = cot−1 x + x increases in the interval
(a) (1, ∞)
(b) (−1, ∞)
(c) (−∞, ∞)
(d) (0, ∞)

(c) (−∞, ∞)

#### Question 3:

The function f(x) = xx decreases on the interval
(a) (0, e)
(b) (0, 1)
(c) (0, 1/e)
(d) none of these

(c) (0, 1/e)

#### Question 4:

The function f(x) = 2 log (x − 2) − x2 + 4x + 1 increases on the interval
(a) (1, 2)
(b) (2, 3)
(c) (1, 3)
(d) (2, 4)

(b) (2, 3)

#### Question 5:

If the function f(x) = 2x2kx + 5 is increasing on [1, 2], then k lies in the interval
(a) (−∞, 4)
(b) (4, ∞)
(c) (−∞, 8)
(d) (8, ∞)

(a) (−∞, 4)

#### Question 6:

Let f(x) = x3 + ax2 + bx + 5 sin2x be an increasing function on the set R. Then, a and b satisfy
(a) a2 − 3b − 15 > 0
(b) a2 − 3b + 15 > 0
(c) a2 − 3b + 15 < 0
(d) a > 0 and b > 0

#### Answer:

(c) a2 − 3b + 15 < 0

#### Question 7:

The function $f\left(x\right)={\mathrm{log}}_{e}\left({x}^{3}+\sqrt{{x}^{6}+1}\right)$ is of the following types:
(a) even and increasing
(b) odd and increasing
(c) even and decreasing
(d) odd and decreasing

#### Answer:

(b) odd and increasing

#### Question 8:

If the function f(x) = 2 tan x + (2a + 1) loge | sec x | + (a − 2) x is increasing on R, then
(a) a ∈ (1/2, ∞)
(b) a ∈ (−1/2, 1/2)
(c) a = 1/2
(d) aR

#### Question 9:

Let $f\left(x\right)={\mathrm{tan}}^{-1}\left(g\left(x\right)\right),$ where g (x) is monotonically increasing for 0 < x < $\frac{\mathrm{\pi }}{2}.$ Then, f(x) is
(a) increasing on (0, π/2)
(b) decreasing on (0, π/2)
(c) increasing on (0, π/4) and decreasing on (π/4, π/2)
(d) none of these

#### Answer:

(a) increasing on (0, $\mathrm{\pi }$/2)

#### Question 10:

Let f(x) = x3 − 6x2 + 15x + 3. Then,
(a) f(x) > 0 for all xR
(b) f(x) > f(x + 1) for all xR
(c) f(x) is invertible
(d) none of these

#### Answer:

(c) f(x) is invertible
f(x) =x3 − 6x2 + 15x + 3

#### Question 11:

The function f(x) = x2 ex is monotonic increasing when
(a) xR − [0, 2]
(b) 0 < x < 2
(c) 2 < x < ∞
(d) x < 0

(b) 0 < x < 2

#### Question 12:

Function f(x) = cos x − 2 λ x is monotonic decreasing when
(a) λ > 1/2
(b) λ < 1/2
(c) λ < 2
(d) λ > 2

(a) λ > 1/2

#### Question 13:

In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
(a) monotonically increasing
(b) monotonically decreasing
(c) not monotonic
(d) constant

#### Answer:

(b) monotonically decreasing

#### Question 14:

Function f(x) = x3 − 27x + 5 is monotonically increasing when
(a) x < −3
(b) | x | > 3
(c) x ≤ −3
(d) | x | ≥ 3

(d) | x | ≥ 3

#### Question 15:

Function f(x) = 2x3 − 9x2 + 12x + 29 is monotonically decreasing when
(a) x < 2
(b) x > 2
(c) x > 3
(d) 1 < x < 2

(d) 1 < x < 2

#### Question 16:

If the function f(x) = kx3 − 9x2 + 9x + 3 is monotonically increasing in every interval, then
(a) k < 3
(b) k ≤ 3
(c) k > 3
(d) k ≥ 3

(c) k > 3

#### Question 17:

f(x) = 2x − tan−1 x − log $\left\{x+\sqrt{{x}^{2}+1}\right\}$ is monotonically increasing when
(a) x > 0
(b) x < 0
(c) xR
(d) xR − {0}

(c) xR

#### Question 18:

Function f(x) = | x | − | x − 1 | is monotonically increasing when
(a) x < 0
(b) x > 1
(c) x < 1
(d) 0 < x < 1

(d) 0 < x < 1

#### Question 19:

Every invertible function is
(a) monotonic function
(b) constant function
(c) identity function
(d) not necessarily monotonic function

#### Answer:

(a) monotonic function

We know that "every invertible function is a monotonic function".

#### Question 20:

In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
(a) increasing
(b) decreasing
(c) constant
(d) none of these

(b) decreasing

#### Question 21:

If the function f(x) = cos |x| − 2ax + b increases along the entire number scale, then

(a) a = b

(b) $a=\frac{1}{2}b$

(c) $a\le -\frac{1}{2}$

(d) $a>-\frac{3}{2}$

#### Answer:

(c) $a\le -\frac{1}{2}$

#### Question 22:

The function
(a) strictly increasing
(b) strictly decreasing
(c) neither increasing nor decreasing
(d) none of these

#### Answer:

(a)  strictly increasing

#### Question 23:

The function is increasing, if
(a) λ < 1
(b) λ > 1
(c) λ < 2
(d) λ > 2

(d) λ > 2

#### Question 24:

Function f(x) = ax is increasing on R, if
(a) a > 0
(b) a < 0
(c) 0 < a < 1
(d) a > 1

(d) a > 1

#### Question 25:

Function f(x) = loga x is increasing on R, if
(a) 0 < a < 1
(b) a > 1
(c) a < 1
(d) a > 0

(b) a > 1

#### Question 26:

Let ϕ(x) = f(x) + f(2ax) and f"(x) > 0 for all x ∈ [0, a]. Then, ϕ (x)
(a) increases on [0, a]
(b) decreases on [0, a]
(c) increases on [−a, 0]
(d) decreases on [a, 2a]

#### Answer:

Given: ϕ(x) = f(x) + f(2ax)

Differentiating above equation with respect to x we get,

ϕ'(x) = f'(x) − f(2ax)        .....(1)

Since, f''(x) > 0, f'(x) is an increasing function.

Now,

when

Considering equation (1) and (2) we get,

ϕ'(x) ≤ 0

⇒ ϕ'(x) is decreasing in [0, a]

#### Question 27:

If the function f(x) = x2kx + 5 is increasing on [2, 4], then
(a) k ∈ (2, ∞)
(b) k ∈ (−∞, 2)
(c) k ∈ (4, ∞)
(d) k ∈ (−∞, 4).

(d) k ∈ (−∞, 4)

#### Question 28:

The function f(x) = −x/2 + sin x defined on [−π/3, π/3] is
(a) increasing
(b) decreasing
(c) constant
(d) none of these

#### Answer:

Hence, the given function is increasing .

#### Question 29:

If the function f(x) = x3 − 9kx2 + 27x + 30 is increasing on R, then
(a) −1 ≤ k < 1
(b) k < −1 or k > 1
(c) 0 < k < 1
(d) −1 < k < 0

(a)

#### Question 30:

The function f(x) = x9 + 3x7 + 64 is increasing on
(a) R
(b) (−∞, 0)
(c) (0, ∞)
(d) R0

(a) R

#### Question 31:

The interval on which the function f(x) = 2x3 + 9x2 + 12x - 1 is decreasing is
(a) [-1, ∞)            (b) [-2, -1]             (c) (∞, -2]               (d) [-1, 1]

#### Answer:

$f\left(x\right)=2{x}^{3}+9{x}^{2}+12x-1$

$⇒f\text{'}\left(x\right)=6{x}^{2}+18x+12$

$⇒f\text{'}\left(x\right)=6\left({x}^{2}+3x+2\right)$

$⇒f\text{'}\left(x\right)=6\left(x+2\right)\left(x+1\right)$

For f(x) to be decreasing,

$f\text{'}\left(x\right)<0$

$⇒6\left(x+2\right)\left(x+1\right)<0$

$⇒\left[x-\left(-2\right)\right]\left[x-\left(-1\right)\right]<0$

$⇒-2         [For a < b, if (xa)(xb) < 0 ⇒ a < x < b]

Thus, the interval on which the given function f(x) is decreasing is [−2, −1].

Hence, the correct answer is option (b).

#### Question 32:

y = x(x-3)2 decrease for the values of x given by
(a) 1 < x < 3                (b) x < 0                (c) x > 0                    (d) 0 < x$\frac{3}{2}$

#### Answer:

yx(x − 3)2

Differentiating both sides with respect to x, we get

$\frac{dy}{dx}=x×2\left(x-3\right)+{\left(x-3\right)}^{2}×1$

$⇒\frac{dy}{dx}=\left(x-3\right)\left(2x+x-3\right)$

$⇒\frac{dy}{dx}=\left(x-3\right)\left(3x-3\right)$

$⇒\frac{dy}{dx}=3\left(x-1\right)\left(x-3\right)$

For y to be decreasing,

$\frac{dy}{dx}<0$

$⇒3\left(x-1\right)\left(x-3\right)<0$

$⇒\left(x-1\right)\left(x-3\right)<0$

$⇒1                  [For a < b, if (x − a)(x − b) < 0 ⇒ axb]

Thus, y decreases for 1 < x < 3.

Hence, the correct answer is option (a).

#### Question 33:

The function f(x) = 4sin3x - 6sin2x + 12sin x + 100 is strictly
(a) increasing in                           (b) decreasing in $\left(\frac{\mathrm{\pi }}{2},\mathrm{\pi }\right)$
(c) decreasing in $\left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right]$                      (d) decreasing in $\left[0,\frac{\mathrm{\pi }}{2}\right]$

#### Answer:

The given function is f(x) = 4sin3x − 6sin2x + 12sinx + 100.

f(x) = 4sin3x − 6sin2x + 12sinx + 100

Differentiating both sides with respect to x, we get

$f\text{'}\left(x\right)=4×3{\mathrm{sin}}^{2}x×\mathrm{cos}x-6×2\mathrm{sin}x×\mathrm{cos}x+12\mathrm{cos}x$

$f\text{'}\left(x\right)=12{\mathrm{sin}}^{2}x\mathrm{cos}x-12\mathrm{sin}x\mathrm{cos}x+12\mathrm{cos}x$

$⇒f\text{'}\left(x\right)=12\mathrm{cos}x\left({\mathrm{sin}}^{2}x-\mathrm{sin}x+1\right)$

$⇒f\text{'}\left(x\right)=12\mathrm{cos}x\left[{\left(\mathrm{sin}x-\frac{1}{2}\right)}^{2}+\frac{3}{4}\right]$

Now,

${\left(\mathrm{sin}x-\frac{1}{2}\right)}^{2}+\frac{3}{4}>0$x ∈ R

When $x\in \left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right]$, cosx ≥ 0

$\therefore f\text{'}\left(x\right)\ge 0$

So, f(x) is increasing in $\left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right]$.

When $x\in \left[0,\frac{\mathrm{\pi }}{2}\right]$, cosx ≥ 0

$\therefore f\text{'}\left(x\right)\ge 0$

So, f(x) is increasing in $\left[0,\frac{\mathrm{\pi }}{2}\right]$.

When , cosx ≤ 0

$\therefore f\text{'}\left(x\right)\le 0$

So, f(x) is decreasing in .

When $x\in \left(\frac{\mathrm{\pi }}{2},\mathrm{\pi }\right)$, cosx < 0

$\therefore f\text{'}\left(x\right)<0$

So, f(x) is strictly decreasing in $\left(\frac{\mathrm{\pi }}{2},\mathrm{\pi }\right)$.

Thus, the function f(x) = 4sin3x − 6sin2x + 12sin x + 100 is strictly decreasing in $\left(\frac{\mathrm{\pi }}{2},\mathrm{\pi }\right)$.

Hence, the correct answer is option (b).

#### Question 34:

Which of the following functions is decreasing in $\left(0,\frac{\mathrm{\pi }}{2}\right)$?
(a) sin 2x          (b) tan x              (c) cos x            (d) cos 3x

#### Answer:

Let f(x) = sin2x

$\therefore f\text{'}\left(x\right)=2\mathrm{cos}2x$

$0    (Given)

$⇒0<2x<\mathrm{\pi }$

Now, cos2x > 0 when $0<2x<\frac{\mathrm{\pi }}{2}$ and cos2x < 0 when $\frac{\mathrm{\pi }}{2}<2x<\mathrm{\pi }$.

$⇒f\text{'}\left(x\right)>0$ when $0 and $f\text{'}\left(x\right)<0$ when $\frac{\mathrm{\pi }}{4}

f(x) is increasing when $0 and f(x) is decreasing when $\frac{\mathrm{\pi }}{4}

Thus, f(x) = sin2x is both increasing and decreasing in the interval $\left(0,\frac{\mathrm{\pi }}{2}\right)$.

Let g(x) = tanx

$\therefore g\text{'}\left(x\right)={\mathrm{sec}}^{2}x$

Now, sec2x > 0 when $0

$⇒g\text{'}\left(x\right)>0$ when $0

⇒ g(x) = tanx is increasing when $0

Let h(x) = cosx

$\therefore h\text{'}\left(x\right)=-\mathrm{sin}x$

Now, sinx > 0 when $0

$⇒h\text{'}\left(x\right)<0$ when $0

⇒ h(x) = cosx is decreasing when $0

Let p(x) = cos3x

$\therefore p\text{'}\left(x\right)=-3\mathrm{sin}3x$

$0    (Given)

$⇒0<3x<\frac{3\mathrm{\pi }}{2}$

Now, sin3x > 0 when $0<3x<\mathrm{\pi }$ and sin3x < 0 when $\mathrm{\pi }<3x<\frac{3\mathrm{\pi }}{2}$.

$⇒p\text{'}\left(x\right)<0$ when $0 and $p\text{'}\left(x\right)>0$ when $\frac{\mathrm{\pi }}{3}

p(x) is decreasing when $0 and p(x) is increasing when $\frac{\mathrm{\pi }}{3}

p(x) = cos3x is both increasing and decreasing in the interval $\left(0,\frac{\mathrm{\pi }}{2}\right)$.

Thus, the function cosx is decreasing in $\left(0,\frac{\mathrm{\pi }}{2}\right)$.

Hence, the correct answer is option (c).

#### Question 35:

The function f(x) = tan x - x
(a) always increases                    (b) always decreases
(c) never increases                      (d) sometimes increases sometime decreases

#### Answer:

The given function is $f\left(x\right)=\mathrm{tan}x-x$.

$f\left(x\right)=\mathrm{tan}x-x$

Differentiating both sides with respect to x, we get

$f\text{'}\left(x\right)={\mathrm{sec}}^{2}x-1$

We know

$\mathrm{sec}x\in \left(-\infty ,-1\right]\cup \left[1,\infty \right)$

$⇒{\mathrm{sec}}^{2}x\in \left[1,\infty \right)$

Or sec2x ≥ 1 for all real values of x

f(x) is increasing for all x ∈ R

Thus, the function f(x) always increases for all real values of x.

Hence, the correct answer is option (a).

#### Question 1:

The values of 'a' for which the function f(x) = sin x ax + b increases on R are _______________.

#### Answer:

The given function is f(x) = sinx ax + b.

f(x) = sinx ax + b

Differentiating both sides with respect to x, we get

$f\text{'}\left(x\right)=\mathrm{cos}x-a$

It is given that, f(x) increases on R.

$\therefore f\text{'}\left(x\right)\ge 0$x ∈ R

$⇒\mathrm{cos}x-a\ge 0$x ∈ R

$⇒a\le \mathrm{cos}x$x ∈ R

a ∈ (−∞, −1]

Thus, the values of 'a' for which the function f(x) = sin x ax + b increases on R are (−∞, −1].

The values of 'a' for which the function f(x) = sin x ax + b increases on R are ___(−∞, −1]___.

#### Question 2:

The function f(x) = $\frac{2{x}^{2}-1}{{x}^{4}},$ x > 0, decreases in the interval ________________.

#### Answer:

The given function is .

$f\left(x\right)=\frac{2{x}^{2}-1}{{x}^{4}}$

$⇒f\left(x\right)=\frac{2}{{x}^{2}}-\frac{1}{{x}^{4}}$

Differentiating both sides with respect to x, we get

$f\text{'}\left(x\right)=-\frac{4}{{x}^{3}}+\frac{4}{{x}^{5}}$

$⇒f\text{'}\left(x\right)=-4\left(\frac{{x}^{2}-1}{{x}^{5}}\right)$

$⇒f\text{'}\left(x\right)=\frac{-4\left(x+1\right)\left(x-1\right)}{{x}^{5}}$

For f(x) to be decreasing,

$f\text{'}\left(x\right)<0$

$⇒\frac{-4\left(x+1\right)\left(x-1\right)}{{x}^{5}}<0$

$⇒\left[x-\left(-1\right)\right]\left(x-1\right)>0$

$⇒x\in \left(-\infty ,-1\right)\cup \left(1,\infty \right)$          [For ab, if (x − a)(x − b) > 0 ⇒ x ∈ (−∞, a) ∪ (b, ∞)]

But, x > 0       (Given)

∴ x ∈ (1, ∞)

Thus, the given function f(x) decreases in the interval (1, ∞).

The function f(x) = $\frac{2{x}^{2}-1}{{x}^{4}},$ x > 0, decreases in the interval ____(1, ∞)____.

#### Question 3:

The function g(x) = x$\frac{1}{x},x\ne 0$ decreases in the closed interval ____________________.

#### Answer:

The given function is .

$g\left(x\right)=x+\frac{1}{x}$

Differentiating both sides with respect to x, we get

$g\text{'}\left(x\right)=1-\frac{1}{{x}^{2}}$

For g(x) to be decreasing,

$g\text{'}\left(x\right)<0$

$⇒1-\frac{1}{{x}^{2}}<0$

$⇒\frac{{x}^{2}-1}{{x}^{2}}<0$

$⇒\left(x+1\right)\left(x-1\right)<0$

$⇒\left[x-\left(-1\right)\right]\left(x-1\right)<0$                 [For ab, if (x − a)(x − b) < 0 ⇒ axb]

$⇒-1

∴ x ∈ (−1, 1)

Thus, the given function g(x) decreases in the interval (−1, 1).

The function g(x) = x + $\frac{1}{x},x\ne 0$ decreases in the closed interval _____[−1, 1]_____.

#### Question 4:

The largest open interval in which the function f(x) = $\frac{1}{1+{x}^{2}}$ decreases is _______________.

#### Answer:

The given function is $f\left(x\right)=\frac{1}{1+{x}^{2}}$.

$f\left(x\right)=\frac{1}{1+{x}^{2}}$

Differentiating both sides with respect to x, we get

$f\text{'}\left(x\right)=-\frac{2x}{{\left(1+{x}^{2}\right)}^{2}}$

For f(x) to be decreasing,

$f\text{'}\left(x\right)<0$

$⇒-\frac{2x}{{\left(1+{x}^{2}\right)}^{2}}<0$

$⇒x>0$

∴ x ∈ (0, ∞)

Thus, the largest open interval in which the given function f(x) decreases is (0, ∞).

The largest open interval in which the function f(x) = $\frac{1}{1+{x}^{2}}$ decreases is ____(0, ∞)____.

#### Question 5:

The set of values of x for which f(x) = tan x - x is increasing is _______________.

#### Answer:

The given function is $f\left(x\right)=\mathrm{tan}x-x$.

$f\left(x\right)=\mathrm{tan}x-x$

Differentiating both sides with respect to x, we get

$f\text{'}\left(x\right)={\mathrm{sec}}^{2}x-1$

For f(x) to be increasing,

$f\text{'}\left(x\right)>0$

$⇒{\mathrm{sec}}^{2}x-1>0$

$⇒{\mathrm{sec}}^{2}x>1$

We know

$\mathrm{sec}x\in \left(-\infty ,-1\right]\cup \left[1,\infty \right)$

$⇒{\mathrm{sec}}^{2}x\in \left[1,\infty \right)$

Or sec2x ≥ 1 for all real values of x

Thus, the set of values of x for which f(x) is increasing is the set of all real numbers i.e. R.

The set of values of x for which f(x) = tan x − x is increasing is ___the set of all real numbers i.e. R___.

#### Question 6:

The set of values of  'a' for which the function f(x) = sin x - cos x - ax + b decreases for all the real values of x, is ___________.

#### Answer:

The given function is f(x) = sin x − cos x ax + b.

f(x) = sin x − cos  ax + b

Differentiating both sides with respect to x, we get

$f\text{'}\left(x\right)=\mathrm{cos}x-\left(-\mathrm{sin}x\right)-a$

$⇒f\text{'}\left(x\right)=\mathrm{cos}x+\mathrm{sin}x-a$

$⇒f\text{'}\left(x\right)=\sqrt{2}\left(\mathrm{cos}x×\frac{1}{\sqrt{2}}+\mathrm{sin}x×\frac{1}{\sqrt{2}}\right)-a$

$⇒f\text{'}\left(x\right)=\sqrt{2}\left(\mathrm{cos}x\mathrm{cos}\frac{\mathrm{\pi }}{4}+\mathrm{sin}x\mathrm{sin}\frac{\mathrm{\pi }}{4}\right)-a$

$⇒f\text{'}\left(x\right)=\sqrt{2}\mathrm{cos}\left(x-\frac{\mathrm{\pi }}{4}\right)-a$

For f(x) to be decreasing for all x,

$f\text{'}\left(x\right)\le 0$

$⇒\sqrt{2}\mathrm{cos}\left(x-\frac{\mathrm{\pi }}{4}\right)-a\le 0$

$⇒a\ge \sqrt{2}\mathrm{cos}\left(x-\frac{\mathrm{\pi }}{4}\right)$

$\therefore a\in \left[\sqrt{2},\infty \right)$

Thus, the set of values of  'a' for which the given function f(x) decreases for all the real values of x is $\left[\sqrt{2},\infty \right)$.

The set of values of  'a' for which the function f(x) = sin x − cos x ax + b decreases for all the real values of x, is .

#### Question 7:

The set of values of  'a' for which the function f(x) = ax + b is strictly increasing for all real x, is _______________.

#### Answer:

The given function is f(x) = ax + b.

f(x) = ax + b

Differentiating both sides with respect to x, we get

$f\text{'}\left(x\right)=a$

For f(x) to be strictly increasing for all real x,

$f\text{'}\left(x\right)>0$

$⇒a>0$

$\therefore a\in \left(0,\infty \right)$

Thus, the set of values of 'a' for which the function f(x) = ax + b is strictly increasing for all real x is (0, ∞).

The set of values of  'a' for which the function f(x) = ax + b is strictly increasing for all real x, is ____(0, ∞)____.

#### Question 8:

If kπ is the length of the largest interval in which the function f(x) = 3sin x - 4sin3x is increasing, then k = _________________.

#### Answer:

The given function is f(x) = 3sin x − 4sin3x.

f(x) = 3sin x − 4sin3x = sin3x

Differentiating both sides with respect to x, we get

$f\text{'}\left(x\right)=3\mathrm{cos}3x$

For f(x) to be increasing,

$f\text{'}\left(x\right)\ge 0$

$⇒3\mathrm{cos}3x\ge 0$

$⇒\mathrm{cos}3x\ge 0$

$⇒-\frac{\mathrm{\pi }}{2}\le 3x\le \frac{\mathrm{\pi }}{2}$

$⇒-\frac{\mathrm{\pi }}{6}\le x\le \frac{\mathrm{\pi }}{6}$

∴ Length of the largest interval in which the given function f(x) is increasing = $\frac{\mathrm{\pi }}{6}-\left(-\frac{\mathrm{\pi }}{6}\right)=\frac{2\mathrm{\pi }}{6}=\frac{\mathrm{\pi }}{3}$

It is given that, the length of the largest interval in which the function f(x) = 3sin x − 4sin3x is increasing is k$\mathrm{\pi }$.

$\therefore k\mathrm{\pi }=\frac{\mathrm{\pi }}{3}$

$⇒k=\frac{1}{3}$

Thus, the value of k is $\frac{1}{3}$.

If kπ is the length of the largest interval in which the function f(x) = 3sin x − 4sin3x is increasing, then k = .

#### Question 9:

The set of values of λ for which the function f(x) = $\frac{\lambda \mathrm{sin}x+6\mathrm{cos}x}{2\mathrm{sin}x+3\mathrm{cos}x}$ is strictly increasing, is ___________________.

#### Answer:

The given function is $f\left(x\right)=\frac{\lambda \mathrm{sin}x+6\mathrm{cos}x}{2\mathrm{sin}x+3\mathrm{cos}x}$.

$f\left(x\right)=\frac{\lambda \mathrm{sin}x+6\mathrm{cos}x}{2\mathrm{sin}x+3\mathrm{cos}x}$

Differentiating both sides with respect to x, we get

$f\text{'}\left(x\right)=\frac{\left(2\mathrm{sin}x+3\mathrm{cos}x\right)×\left(\lambda \mathrm{cos}x-6\mathrm{sin}x\right)-\left(\lambda \mathrm{sin}x+6\mathrm{cos}x\right)×\left(2\mathrm{cos}x-3\mathrm{sin}x\right)}{{\left(2\mathrm{sin}x+3\mathrm{cos}x\right)}^{2}}$

$⇒f\text{'}\left(x\right)=\frac{2\lambda \mathrm{sin}x\mathrm{cos}x-12{\mathrm{sin}}^{2}x+3\lambda {\mathrm{cos}}^{2}x-18\mathrm{sin}x\mathrm{cos}x-2\lambda \mathrm{sin}x\mathrm{cos}x+3\lambda {\mathrm{sin}}^{2}x-12{\mathrm{cos}}^{2}x+18\mathrm{sin}x\mathrm{cos}x}{{\left(2\mathrm{sin}x+3\mathrm{cos}x\right)}^{2}}$

$⇒f\text{'}\left(x\right)=\frac{3\lambda \left({\mathrm{sin}}^{2}x+{\mathrm{cos}}^{2}x\right)-12\left({\mathrm{sin}}^{2}x+{\mathrm{cos}}^{2}x\right)}{{\left(2\mathrm{sin}x+3\mathrm{cos}x\right)}^{2}}$

$⇒f\text{'}\left(x\right)=\frac{3\lambda -12}{{\left(2\mathrm{sin}x+3\mathrm{cos}x\right)}^{2}}$         [sin2x + cos2x = 1]

For f(x) to be strictly increasing,

$f\text{'}\left(x\right)>0$

$⇒\frac{3\lambda -12}{{\left(2\mathrm{sin}x+3\mathrm{cos}x\right)}^{2}}>0$

$⇒3\lambda >12$

$⇒\lambda >4$

Thus, the set of values of λ for which the function f(x) is strictly increasing is (4, ∞).

The set of values of λ for which the function f(x) = $\frac{\lambda \mathrm{sin}x+6\mathrm{cos}x}{2\mathrm{sin}x+3\mathrm{cos}x}$ is strictly increasing, is ____(4, ∞)____.

#### Question 10:

The largest interval in which f(x) = x1/x is strictly increasing is ______________.

#### Answer:

The given function is $f\left(x\right)={x}^{\frac{1}{x}}$.

For f(x) to be defined x > 0.

$f\left(x\right)={x}^{\frac{1}{x}}$

$⇒\mathrm{log}f\left(x\right)=\mathrm{log}{x}^{\frac{1}{x}}$

Differentiating both sides with respect to x, we get

$\frac{1}{f\left(x\right)}×f\text{'}\left(x\right)=\frac{x×\frac{1}{x}-\mathrm{log}x×1}{{x}^{2}}$

$⇒f\text{'}\left(x\right)=\frac{{x}^{\frac{1}{x}}\left(1-\mathrm{log}x\right)}{{x}^{2}}$

For f(x) to be strictly increasing function,

$f\text{'}\left(x\right)>0$

$⇒\frac{{x}^{\frac{1}{x}}\left(1-\mathrm{log}x\right)}{{x}^{2}}>0$

$⇒\mathrm{log}x<1$

$⇒\mathrm{log}x<\mathrm{log}e$

$⇒x

$⇒x\in \left(0,e\right)$         (x > 0)

Thus, the largest interval in which f(x) = x1/x is strictly increasing is (0, e).

The largest interval in which f(x) = x1/x is strictly increasing is ____(0, e)____.

#### Question 1:

What are the values of 'a' for which f(x) = ax is increasing on R?

#### Question 2:

What are the values of 'a' for which f(x) = ax is decreasing on R?

#### Question 3:

Write the set of values of 'a' for which f(x) = loga x is increasing in its domain.

#### Question 4:

Write the set of values of 'a' for which f(x) = loga x is decreasing in its domain.

#### Question 5:

Find 'a' for which f(x) = a (x + sin x) + a is increasing on R.

#### Question 6:

Find the values of 'a' for which the function f(x) = sin xax + 4 is increasing function on R.

#### Question 7:

Find the set of values of 'b' for which f(x) = b (x + cos x) + 4 is decreasing on R.

#### Question 8:

Find the set of values of 'a' for which f(x) = x + cos x + ax + b is increasing on R.

#### Question 9:

Write the set of values of k for which f(x) = kx − sin x is increasing on R.

#### Question 10:

If g (x) is a decreasing function on R and f(x) = tan−1 [g (x)]. State whether f(x) is increasing or decreasing on R.

#### Question 11:

Write the set of values of a for which the function f(x) = ax + b is decreasing for all xR.

#### Question 12:

Write the interval in which f(x) = sin x + cos x, x ∈ [0, π/2] is increasing.

#### Question 13:

State whether f(x) = tan xx is increasing or decreasing its domain.

#### Question 14:

Write the set of values of a for which f(x) = cos x + a2 x + b is strictly increasing on R.

#### Answer:

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