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

on the closed interval

[0,2]

. (1) Show that

f

satisfies the necessary continuity and differentiability conditions on

[0,2]

so that a value

c

with

f'(c)=\dfrac{f(2)-f(0)}{2-0}

is guaranteed to exist. (2) Calculate the average rate of change of

f

[0,2]

, giving an exact value. (3) Determine every number

c

in the open interval

(0,2)

such that

f'(c)

equals the average rate of change found in part (b). (4) Verify explicitly that each value of

c

found in part (c) lies in

(0,2)