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Question Number 12551 by @ANTARES_VY last updated on 25/Apr/17

$$\mathbf{This}\mathbf{y}=\frac{2{\mathbf{cos}}^2\mathbf{x}+\mathbf{sin}2\mathbf{x}}{2{\mathbf{sin}}^2\mathbf{x}}\mathbf{find}\mathbf{the}$$$$\mathbf{smallest}\mathbf{value}\mathbf{of}\mathbf{the}\mathbf{function}.$$

Answered by mrW1 last updated on 25/Apr/17

$$\mathbf{y}=\frac{2{\mathbf{cos}}^2\mathbf{x}+\mathbf{sin}2\mathbf{x}}{2{\mathbf{sin}}^2\mathbf{x}}$$$$=\frac{2{\mathbf{cos}}^2\mathbf{x}+2\mathbf{sinx}\cos x}{2{\mathbf{sin}}^2\mathbf{x}}$$$$=\frac{{\mathbf{cos}}^2\mathbf{x}+\mathbf{sinx}\cos x}{{\mathbf{sin}}^2\mathbf{x}}$$$$={\cot }^{2}x+\cot x$$$$={\cot }^{2}x+2\times \frac{1}{2}\cot x+{\left(\frac{1}{2}\right)}^2-\frac{1}{4}$$$$={(\cot x+\frac{1}{2})}^2-\frac{1}{4}⩾-\frac{1}{4}$$$$\Rightarrow smallesvalueoffunction=-\frac{1}{4}$$$$$$$$minimumwhen\cot x+\frac{1}{2}=0$$$$or\tan x=-2$$$$orx=n\pi -{\tan }^{-1}\left(2\right)$$

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