Let be defined for all real .

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

Let

g(x)=x^{4}e^{-x}

be defined for all real

x

. (1) Compute

g'(x)

and determine all critical points of

g

. (2) Compute

g''(x)

and evaluate

g''

at each critical point found in part (a). (3) Classify each critical point of

g

as a relative maximum, relative minimum, or neither. Provide complete justification, especially in any case where your numerical value of

g''

does not immediately decide the classification. (4) Confirm your classifications in part (c) by analyzing the sign of

g'(x)

on appropriate open intervals. Summarize your findings regarding the local extrema of

g

. (A sign chart or clearly reasoned narrative is acceptable.)