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Q.
Let $f: R \rightarrow R$ be defined as $f(x)=7 e^{\sin ^2 x}-e^{\cos ^2 x}+2$, then the value of $\sqrt{7 f_{\min }+f_{\max }}$ is equal to
Application of Derivatives
Solution:
$f^{\prime}(x)=\left(7 e^{\sin ^2 x}+e^{\cos ^2 x}\right) \sin 2 x$ and $f(0)=9-e$
$f$ is increase in $\left(0, \frac{\pi}{2}\right) f \left(\frac{\pi}{2}\right)=7 e +11$
$f$ is decrease in $\left(\frac{\pi}{2}, \pi\right) f (\pi)=9- e$
$\therefore \sqrt{7 f _{\min }+ f _{\max }}=\sqrt{7(9- e )+7 e +1}=8$