Let be defined on the closed interval .

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

Let

f(x)=x^{3}-6x^{2}+9x+5

be defined on the closed interval

[0,4]

. (i) Find

f'(x)

and hence obtain all critical points that lie inside the interval

[0,4]

. (ii) Classify each critical point obtained in part (i) as a local maximum or a local minimum. (iii) Evaluate

f(x)

at every critical point and at the endpoints

x=0

and

x=4

. Hence determine the absolute maximum and absolute minimum values of

f

on the interval

[0,4]

, stating the points at which they occur.