Mathematics

Problem

Consider the polynomial function

f(x)=x^{3}-6x^{2}+9x+4

defined on the closed interval

[0,5]

. For your calculations you may use the derivative

f'(x)=3x^{2}-12x+9

. a. (i) Find all values of

x

(0,5)

that satisfy

f'(x)=0

. b. (ii) Evaluate

f(x)

at each point obtained in part (i) as well as at the endpoints

x=0

and

x=5

. c. (iii) Determine the absolute maximum value and the absolute minimum value of

f

[0,5]

, stating the points at which these values occur.

Intuitive explanation

This problem is about finding extreme values of a cubic function on a bounded interval—a classic application of calculus. The key insight is that the maximum and minimum must occur either at critical points (where $f'(x)=0$ ) or at the endpoints. Factor the derivative to find where the function's slope is zero, evaluate $f$ at those points and at $x=0$ and $x=5$ , then compare all candidates to identify the global extrema on $[0,5]$ .

Intuitive explanation

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