Question

Let f(x) = $7e^{{\sin }^{2}x}-e^{{\cos }^{2}x}+2,$ then the value of $\sqrt{7f_{min}+f_{max},}$ is________

Answer 1

$f\left(t\right)=7t-\frac{e}{t}+2$ for $t\in [1,e]$, where $t=e^{{\sin }^{2}x}$
$\Rightarrow f^{\prime }\left(t\right)=7+\frac{e}{t^2}>0\forall t\in R$
Hence $f$ is increasing function.
$\therefore f_{min}=f\left(1\right)=7-e+2=9-e$
and $f_{max}=f\left(e\right)=7e-1+2=7e+1$
Hence answer is $8$.