Let be the polynomial function defined by on the closed interval .

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

Problem

Let

f

be the polynomial function defined by

f(x)=x^{3}-3x^{2}-9x+7

on the closed interval

[-2,5]

. (a) Explain why the function

f

is guaranteed to achieve both an absolute maximum value and an absolute minimum value on the interval

[-2,5]

. (b) Compute

f'(x)

and determine all interior

x

–values in

(-2,5)

at which

f'(x)=0

. (c) Evaluate

f

at each

x

found in part (b) and at the endpoints

x=-2

and

x=5

. (d) Identify the absolute maximum value and the absolute minimum value of

f

[-2,5]

, stating the

x

–location and corresponding function value for each and providing justification for your choice.