If f(x)= 7 e^ ^2 x - e^ ^2 x + 2, then, 7 f_ + f_ is equal to - Vidyadip

MCQ

If $f(x)=$ $7{e^{{{\sin }^2}x}} - {e^{{{\cos }^2}x}} + 2$, then,$\sqrt {7{f_{\min }} + {f_{\max }}}$ is equal to

A
$0$
B
$\sqrt {10}$
C
$\sqrt {8}$
✓
$8$

✓

Answer

Correct option: D.

$8$

d
$f(t)=7 t-\frac{e}{t}+2$ for $t \in[1, e],$ where $t=e^{\sin ^{2} x}$

$\Rightarrow f^{\prime}(t)=7+\frac{e}{t^{2}}>0 \forall t \in R$

Hence $f$ is increasing function.

$\therefore f_{\min }=f(1)=7-e+2=9-e$

and $f_{\max }=f(e)=7 e-1+2=7 e+1$

Hence answer is 8 .

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