The maximum value of x^4 e^ - x^2 is - Vidyadip

MCQ

The maximum value of ${x^4}{e^{ - {x^2}}}$ is

A
${e^2}$
B
${e^{ - 2}}$
C
$12{e^{ - 2}}$
✓
$4{e^{ - 2}}$

✓

Answer

Correct option: D.

$4{e^{ - 2}}$

d
(d) $f(x) = {x^4}{e^{ - {x^2}}}$ ==> $f'(x) = 4{x^3}{e^{ - {x^2}}} + {x^4}{e^{ - {x^2}}}( - 2x)$

For max., $f'(x) = 0$ ==> $4{x^3}{e^{ - {x^2}}} - 2{x^5}{e^{ - {x^2}}} = 0$

==> ${x^2} = 2 \Rightarrow x = \pm \sqrt 2 $

$f '' (x) = 12 x^2 e^{-x{^2}} + 4x^3 e^{-x{^2}} (-2x) -10 x^4 e^{-x{^2}} -2x^5 e^{-x{^2}} (-2x)$

==> $f''(\sqrt 2 ) = 24{e^{ - 2}} - 32{e^{ - 2}} - 40{e^{ - 2}} + 32{e^{ - 2}}$ = -ve

Hence, $f(x)$ is max. at $x = \sqrt 2 $

$\therefore$ Maximum value = $4{e^{ - 2}}$.

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