In ( - 4, ,4) the function f(x) = _ - 10 ^x ( t^4 - 4) e^ - 4t dt… - Vidyadip

MCQ

In $( - 4,\,4)$ the function $f(x) = \int\limits_{ - 10}^x {({t^4} - 4){e^{ - 4t}}dt} $ has

A
No extrema
B
One extremum
✓
Two extrema
D
Four extrema

✓

Answer

Correct option: C.

Two extrema

c
(c) $f(x) = \int_{ - 10}^x {({t^4} - 4){e^{ - 4t}}dt} $ ==> $f'(x) = ({x^4} - 4){e^{ - 4x}}$

Now $f'(x) = 0 \Rightarrow x = \pm \sqrt 2 ,\, \pm \sqrt 2 $

Now $f''(x) = - \,4({x^4} - 4){e^{ - 4x}} + 4{x^3}{e^{ - 4x}}$

At $x = \sqrt 2 $ and $x = - \sqrt 2 $ the given function has extreme value.

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