If function f(x) = x(x + 3) e^{-x/2} ; satisfies rolle's theorem in interval [-3, 0], then find

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5. Continuity and Differentiation

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If function $f(x) = x(x + 3) e^{-x/2} ;$ satisfies the rolle's theorem in the interval $[-3, 0],$ then find $C$

A

$0$

B

$1$

C

$-2$

D

$-3$

Solution

$f(x)=\left(x^{2}+3 x\right) e^{-x / 2}$

$f^{\prime}(x)=(2 x+3) e^{-x / 2}-\frac{1}{2}\left(x^{2}+3 x\right) e^{-x / 2}$

$(2 x+3)-\frac{\left(x^{2}+3 x\right)}{2}=0$

$4 x+6-x^{2}-3 x=0$

$x^{2}-x-6=0$

$(x-3)(x+2)=0$

$x=-2,3$

Standard 12

Mathematics

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