1. Tardigrade
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  4. Let f ( x )= x 3-4 x 2-16 x +64. If ∫ limitsα4 f ( x ) dx is maximum, where α<0, then α is equal to

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Q. Let $f ( x )= x ^3-4 x ^2-16 x +64$. If $\int\limits_\alpha^4 f ( x ) dx$ is maximum, where $\alpha<0$, then $\alpha$ is equal to

Application of Derivatives

Solution:

image
$ f(x)=x^2(x-4)-16(x-4) $
$=\left(x^2-16\right)(x-4) $
$=(x-4)^2(x+4)$
For $\int\limits_\alpha^4 f(x) d x$ to be maximum, $\alpha$ must be -4 .