Consider the function defined analytically by

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

Problem

f(x)=x^{\tfrac{4}{3}}\,(6-x)

, for \,

-1\le x \le 6.

Throughout this problem, answers should be exact unless otherwise stated. (a) Calculate

f'(x)

and state its domain. (b) Determine all values of

x

in the interval

[-1,6]

for which

f'(x)=0.

x

in the interval

[-1,6]

for which

f'(x)

does not exist. Explain why each such value is in the domain of

f

. (d) Using your work in parts (b) and (c), list every critical point of

f

[-1,6]

. (e) Evaluate

f

at each critical point and at the endpoints to identify the absolute maximum and absolute minimum values of

f

[-1,6]

. Justify your conclusion.