Let be defined for .

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

Problem

Let

f(x)=x^{3}-6x^{2}+9x+2

be defined for

-1\le x\le 5

. (a) Calculate

f'(x)

. (b) Solve

f'(x)=0

and list all critical numbers that lie in the open interval

(-1,5)

. (c) Compute

f(x)

at every critical number found in part (b) and at the endpoints

x=-1

and

x=5

. (d) Determine the absolute maximum and absolute minimum values of

f

[-1,5]

, stating both the extreme value and the

x

-coordinate at which each occurs. Justify your conclusion with the work from parts (a)–(c).