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Question Number 161590 by geron last updated on 20/Dec/21

$$f\left(x\right)=\frac{1}{1-\sqrt{x}}+\frac{1}{1-\sqrt{1-\mathit{x}}}$$$$\mathit{f}{\left(\mathit{x}\right)}_{\mathit{m}\mathit{i}\mathit{n}}=?$$

Answered by cortano last updated on 20/Dec/21

Answered by mr W last updated on 20/Dec/21

$$0<x<1$$$$letx={\sin }^2(45°+\theta )with-45°<\theta <45°$$$$f\left(x\right)=\frac{1}{1-\sin (45°+\theta )}+\frac{1}{1-\cos (45°+\theta )}$$$$=\frac{1}{1-\sin (45°+\theta )}+\frac{1}{1-\sin (45°-\theta )}$$$$f{\left(x\right)}_{min}=2\times \frac{1}{1-\frac{1}{\sqrt{2}}}=2(2+\sqrt{2})$$$$at\theta =0°orx=\frac{1}{2}$$

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