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Question Number 170155 by cortano1 last updated on 17/May/22

$$Givenf\left(x\right)=x\sqrt{1-x+\sqrt{1-x}}$$$$where0⩽x⩽1$$$$findmaxf\left(x\right)$$

Answered by Mathspace last updated on 18/May/22

$$x\in [0,1]\Rightarrow x=cos\theta \Rightarrow$$$$f\left(x\right)=cos\theta \sqrt{1-cos\theta +\sqrt{1-cos\theta }}$$$$=cos\theta \sqrt{2{sin}^2\left(\frac{\theta }{2}\right)+\sqrt{2{sin}^2\frac{\theta }{2}}}$$$$=cos\theta \sqrt{2{sin}^2\left(\frac{\theta }{2}\right)+\sqrt{2}sin\left(\frac{\theta }{2}\right)}$$$$=cos\theta \sqrt{\sqrt{2}sin\left(\frac{\theta }{2}\right)}\sqrt{\sqrt{2}sin\left(\frac{\theta }{2}\right)+1}$$$$⩽\left({}^4\sqrt{2}\right)\sqrt{1+\sqrt{2}}$$$$....$$

Commented by cortano1 last updated on 18/May/22

$$wrong$$

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