Question

Let $f:R\to R$ be defined as $f(x)=7e^{{\sin }^{2}x}-e^{{\cos }^{2}x}+2$, then the value of $\sqrt{7f_{\min }+f_{\max }}$ is equal to

• A. 2
• B. 4
• C. 6
• D. 8

Answer 1

Answer: (D)

$f^{\prime }(x)=\left(7e^{{\sin }^{2}x}+e^{{\cos }^{2}x}\right)\sin 2x$ and $f(0)=9-e$
$f$ is increase in $\left(0,\frac{\pi }{2}\right)f\left(\frac{\pi }{2}\right)=7e+11$
$f$ is decrease in $\left(\frac{\pi }{2},\pi \right)f(\pi )=9-e$
$\therefore \sqrt{7f_{\min }+f_{\max }}=\sqrt{7(9-e)+7e+1}=8$