Consider the polynomial function on the closed interval

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

Problem

Consider the polynomial function

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

on the closed interval

[0,3]

, where

k\in\mathbb{R}

. (i) Obtain

f'(x)

. (ii) Derive the inequalities that

k

must satisfy so that

f'(x)\ge 0

for every

x\in[0,3]

. (iii) Hence find the complete range of

k

for which

f

is strictly increasing throughout

[0,3]