find intervals in which the function given by f(x) = sin3x, x€[0,π/2] is increasing and decreasing
I know that in the answer the intervals will be [0,π/6) and (π/6,π/2]
But I'm not understanding that how will identify to be increasing or decreasing.

Dear Student,
Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{We} \mathrm{have}, \\ \mathrm{f}\left(\mathrm{x}\right) = \sin 3\mathrm{x} \\ \Rightarrow {\mathrm{f}}^{\text{'}}\left(\mathrm{x}\right) = 3 \cos 3\mathrm{x} \\ \mathrm{Now}, {\mathrm{f}}^{\text{'}}\left(\mathrm{x}\right) = 0 \\ \Rightarrow 3 \cos 3\mathrm{x} = 0 \\ \Rightarrow \cos 3\mathrm{x} = 0 \\ \Rightarrow \cos 3\mathrm{x} = \cos \left(\frac{\mathrm{\pi }}{2}\right) \mathrm{and} \cos 3\mathrm{x} = \cos \left(\frac{3\mathrm{\pi }}{2}\right) \\ \Rightarrow 3\mathrm{x} = \frac{\mathrm{\pi }}{2} \mathrm{or} 3\mathrm{x} = \frac{3\mathrm{\pi }}{2} \\ \Rightarrow \mathrm{x} = \frac{\mathrm{\pi }}{6} \mathrm{or} \mathrm{x} =\frac{\mathrm{\pi }}{2} \\ \mathrm{Now}, \mathrm{x} = \frac{\mathrm{\pi }}{6} \mathrm{divides} \mathrm{the} \mathrm{interval} \left[0, \frac{\mathrm{\pi }}{2}\right] \mathrm{into} 2 \mathrm{disjoint} \mathrm{intervals} : \\ [0, \mathrm{\pi }/6) \mathrm{and} (\mathrm{\pi }/6, \mathrm{\pi }/2]. \\ \mathrm{Consider} \mathrm{the} \mathrm{interval} 0\le \mathrm{x}<\frac{\mathrm{\pi }}{6}. \\ \mathrm{When} 0\le \mathrm{x}<\frac{\mathrm{\pi }}{6} \\ \Rightarrow 3\times 0\le 3\mathrm{x}<3\times \frac{\mathrm{\pi }}{6} \\ \Rightarrow 0\le 3\mathrm{x}<\frac{\mathrm{\pi }}{2} \\ \mathrm{Now}, \cos 3\mathrm{x} > 0 , \mathrm{when} 0\le 3\mathrm{x}<\frac{\mathrm{\pi }}{2} \\ \Rightarrow 3 \cos 3\mathrm{x} > 0, \mathrm{when} 0\le 3\mathrm{x}<\frac{\mathrm{\pi }}{2} \\ \Rightarrow {\mathrm{f}}^{\text{'}}\left(\mathrm{x}\right) > 0, \mathrm{when} 0\le 3\mathrm{x}<\frac{\mathrm{\pi }}{2} \mathrm{or} 0\le \mathrm{x}<\frac{\mathrm{\pi }}{6} \\ \mathbf{So}\mathbf{,}\mathbf{ }\mathbf{f}\left(\mathbf{x}\right)\mathbf{ }\mathbf{is}\mathbf{ }\mathbf{increasing}\mathbf{ }\mathbf{in}\mathbf{ }\mathbf{the}\mathbf{ }\mathbf{interval}\mathbf{ }\mathbf{[}\mathbf{0}\mathbf{,}\mathbf{ }\mathbf{\pi }\mathbf{/}\mathbf{6}\mathbf{)}\mathbf{.} \\ \mathrm{Consider} \mathrm{the} \mathrm{interval} \frac{\mathrm{\pi }}{6}<\mathrm{x}\le \frac{\mathrm{\pi }}{2}. \\ \mathrm{When} \frac{\mathrm{\pi }}{6}<\mathrm{x}\le \frac{\mathrm{\pi }}{2} \\ \Rightarrow 3\times \frac{\mathrm{\pi }}{6}\le 3\mathrm{x}<3\times \frac{\mathrm{\pi }}{2} \\ \Rightarrow \frac{\mathrm{\pi }}{2}<3\mathrm{x}\le \frac{3\mathrm{\pi }}{2} \\ \mathrm{Now}, \cos 3\mathrm{x} < 0 , \mathrm{when} \frac{\mathrm{\pi }}{2}<3\mathrm{x}\le \frac{3\mathrm{\pi }}{2} \\ \Rightarrow 3 \cos 3\mathrm{x} < 0, \mathrm{when} \frac{\mathrm{\pi }}{2}<3\mathrm{x}\le \frac{3\mathrm{\pi }}{2} \\ \Rightarrow {\mathrm{f}}^{\text{'}}\left(\mathrm{x}\right) < 0, \mathrm{when} \frac{\mathrm{\pi }}{2}<3\mathrm{x}\le \frac{3\mathrm{\pi }}{2} \mathrm{or} \frac{\mathrm{\pi }}{6}<\mathrm{x}\le \frac{\mathrm{\pi }}{2} \\ \mathbf{So}\mathbf{,}\mathbf{ }\mathbf{f}\left(\mathbf{x}\right)\mathbf{ }\mathbf{is}\mathbf{ }\mathbf{decreasing}\mathbf{ }\mathbf{in}\mathbf{ }\mathbf{the}\mathbf{ }\mathbf{interval}\mathbf{ }\mathbf{(}\mathbf{\pi }\mathbf{/}\mathbf{6}\mathbf{,}\mathbf{ }\mathbf{\pi }\mathbf{/}\mathbf{2}\mathbf{]}\mathbf{.} \end{gathered}$$



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