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Question Number 63536 by Rio Michael last updated on 05/Jul/19

a) i f y = x^{m} {(1 - x)}^{n}, w h e r e n \in Z^{+}, t h e s e t o f p o s i t i v e i n t e g e r s,

s h o w t h a t w h e n \frac{d y}{d x} = 0, x = \frac{m}{m + n}

b) i f y = 2 (x - 5) \sqrt{x + 4}, s h o w t h a t \frac{d y}{d x} = \frac{3 (x + 1)}{\sqrt{x + 4}}

c) s o l v e t h e e q u a t i o n s i n x - s i n 5 x + c o s 3 x = 0 f o r 0 ° ⩽ x ⩽ 180 °

Commented by Prithwish sen last updated on 05/Jul/19

a . \frac{dy}{dx} = {mx}^{m - 1} {(1 - x)}^{n} - {nx}^{m} {(1 - x)}^{n - 1}

\Rightarrow x^{m - 1} {(1 - x)}^{n - 1} [m (1 - x) - nx] = 0

\Rightarrow m (1 - x) = nx \Rightarrow x = \frac{m}{m + n}

b . \frac{dy}{dx} = 2 \sqrt{x + 4} + \frac{2 (x - 5)}{2 \sqrt{x + 4}} = \frac{3 (x + 1)}{\sqrt{x + 4}}

c . - 2 \cos 3 xsin 2 x + \cos 3 x = 0

either \cos 3 x = 0 or \sin 2 x = \frac{1}{2}

\Rightarrow x = 30^{°} \Rightarrow x = 15^{°}

please check .