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Question Number 132474 by bemath last updated on 14/Feb/21

$$\mathrm{maximum}\mathrm{value}\mathrm{of}$$$$\mathrm{f}\left(\mathrm{x}\right)=\sqrt{4\sin \left(\mathrm{x}\right)+1}-\cos \left(\mathrm{x}\right)$$$$\mathrm{is}$$

Commented by EDWIN88 last updated on 14/Feb/21

Commented by bemath last updated on 14/Feb/21

$$\mathrm{approximation}$$

Answered by mr W last updated on 14/Feb/21

$$\frac{df\left(x\right)}{dx}=\frac{2\cos x}{\sqrt{4\sin x+1}}+\sin x=0$$$$\frac{4{\cos }^{2}x}{4\sin x+1}={\sin }^{2}x$$$$\frac{4(1-{\sin }^{2}x)}{4\sin x+1}={\sin }^{2}x$$$${4\sin }^{3}x+5{\sin }^{2}x-4=0$$$$\frac{1}{{\sin }^{3}x}-\frac{5}{4\sin x}-1=0$$$$\mathrm{\Delta }={(-\frac{1}{2})}^2+{(-\frac{5}{12})}^3=\frac{307}{1728}>0$$$$\Rightarrow \frac{1}{\sin x}=\frac{1}{2}(\sqrt[3]{\frac{\sqrt{921}}{9}+4}-\sqrt[3]{\frac{\sqrt{921}}{9}-4})$$$$\Rightarrow \sin x=\frac{2}{\sqrt[3]{\frac{\sqrt{921}}{9}+4}-\sqrt[3]{\frac{\sqrt{921}}{9}-4}}$$$$\cos x=\pm \sqrt{1-{\sin }^{2}x}$$$$f{\left(x\right)}_{max}=\sqrt{4\sin x+1}+\sqrt{1-{\sin }^{2}x}\approx 2.663815973$$

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