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

$$a)ify=x^m{(1-x)}^n,wheren\in {\mathbb{Z}}^{+},thesetofpositiveintegers,$$$$showthatwhen\frac{dy}{dx}=0,x=\frac{m}{m+n}$$$$b)ify=2(x-5)\sqrt{x+4},showthat\frac{dy}{dx}=\frac{3(x+1)}{\sqrt{x+4}}$$$$c)solvetheequationsinx-sin5x+cos3x=0for0°⩽x⩽180°$$

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

$$\mathbf{a}.\frac{\mathrm{dy}}{\mathrm{dx}}={\mathrm{mx}}^{\mathrm{m}-1}{(1-\mathrm{x})}^{\mathrm{n}}-{\mathrm{nx}}^{\mathrm{m}}{(1-\mathrm{x})}^{\mathrm{n}-1}$$$$\Rightarrow {\mathrm{x}}^{\mathrm{m}-1}{(1-\mathrm{x})}^{\mathrm{n}-1}[\mathrm{m}(1-\mathrm{x})-\mathrm{nx}]=0$$$$\Rightarrow \mathrm{m}(1-\mathrm{x})=\mathrm{nx}\Rightarrow \mathrm{x}=\frac{\mathrm{m}}{\mathrm{m}+\mathrm{n}}$$$$\mathbf{b}.\frac{\mathrm{dy}}{\mathrm{dx}}=2\sqrt{\mathrm{x}+4}+\frac{2(\mathrm{x}-5)}{2\sqrt{\mathrm{x}+4}}=\frac{3(\mathrm{x}+1)}{\sqrt{\mathrm{x}+4}}$$$$\mathbf{c}.-2\cos 3\mathrm{xsin}2\mathrm{x}+\cos 3\mathrm{x}=0$$$$\mathrm{either}\cos 3\mathrm{x}=0\mathrm{or}\sin 2\mathrm{x}=\frac{1}{2}$$$$\Rightarrow \mathrm{x}={30}^{°}\Rightarrow \mathrm{x}={15}^{°}$$$$\mathrm{please}\mathrm{check}.$$

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