Let , defined on the closed interval…

MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives

Problem

Let

g(x)=xe^{-x^{2}}

, defined on the closed interval

[-2,2]

. (a) Determine all values of

x

(-2,2)

at which

g'(x)=0

. (b) For each value found in part (a), compute

g''(x)

and use this information to decide whether

g

has a local maximum, a local minimum, or neither at that point. Justify each answer. (c) Evaluate

g(-2)

and

g(2)

. Using the interior critical points together with the endpoint values, identify the absolute maximum and the absolute minimum of

g

[-2,2]

. Explain your reasoning.