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  4. The function f(x)=x(x+3) e-x / 2 satisfies all the conditions of Rolle's Theorem on [-3,0] . The value of c which verifies Rolle's Theorem, is

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Q. The function $f(x)=x(x+3) e^{-x / 2}$ satisfies all the conditions of Rolle's Theorem on $[-3,0] .$ The value of $c$ which verifies Rolle's Theorem, is

Application of Derivatives

Solution:

$f'(x)=0$
$\Rightarrow (x+3) e^{-x / 2}+x e^{-x / 2}-(x / 2)(x+3) e^{-x / 2}=0$
$\Rightarrow x^{2}-x-6=0$
$\Rightarrow x=-2,3$
$\Rightarrow c=-2 \in(-3,0)$