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Question Number 169826 by cortano1 last updated on 10/May/22

G i v e n f (x) = \sqrt{\sin x} + \sqrt{3 \cos x}

x \in (0, \frac{π}{2})

F i n d m a x f (x) .

Answered by mr W last updated on 10/May/22

f^{'} (x) = \frac{\cos x}{2 \sqrt{\sin x}} - \frac{3 \sin x}{2 \sqrt{3 \cos x}} = 0

{(\tan x)}^{\frac{3}{2}} = \frac{1}{\sqrt{3}}

\Rightarrow \tan x = \frac{1}{\sqrt[3]{3}}

\sin x = \frac{1}{\sqrt{1 + \sqrt[3]{9}}}, \cos x = \frac{\sqrt[3]{3}}{\sqrt{1 + \sqrt[3]{9}}}

f {(x)}_{m a x} = \frac{1 + \sqrt[3]{9}}{\sqrt[4]{1 + \sqrt[3]{9}}} = {(1 + \sqrt[3]{9})}^{\frac{3}{4}} ✓

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