Dear Student, Let C be the total cost FunctionAC = Cx ⇒ C = x×AC = xx+5+36x = x2+5x+36Let MC be the marginal cost function MC = dCdx = ddxx2+5x+36 = 2x + 5For AC to be increasing ddxAC > 0 ⇒ 1 - 36x2 > 0⇒ x2 - 36 > 0⇒ x-6x+6 > 0⇒ x 6As x which is output cannot be negative Therefore, x > 6 Regards,

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Dear Student,

L e t C b e t h e t o t a l \cos t F u n c t i o n A C = \frac{C}{x} \Rightarrow C = x \times A C = x (x + 5 + \frac{36}{x}) = x^{2} + 5 x + 36 L e t M C b e t h e m a r g i n a l \cos t f u n c t i o n . M C = \frac{d C}{d x} = \frac{d}{d x} (x^{2} + 5 x + 36) = 2 x + 5 F o r A C t o b e i n c r e a \sin g \frac{d}{d x} (A C) > 0 \Rightarrow 1 - \frac{36}{x^{2}} > 0 \Rightarrow x^{2} - 36 > 0 \Rightarrow (x - 6) (x + 6) > 0 \Rightarrow x < - 6 o r x > 6 A s x w h i c h i s o u t p u t c a n n o t b e n e g a t i v e . T h e r e f o r e, x > 6

Regards,

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