For positive constants and

MATHEMATICS · Multi-part · A Level · Mathematics · Algebra and Functions · Algebra and Functions

Problem

For positive constants

a

and

b

, consider the function

f_{a,b}(x)=ax^{3}+bx

defined for all real

x

. Let

g_{a,b}=f_{a,b}^{-1}

denote its inverse. (a) Show that

f_{a,b}

is strictly increasing on

\mathbb{R}

. (b) Derive an explicit expression for

g_{a,b}'(x)

in terms of

a

b

, and

x

. (c) When

a=1

and

b=2

, evaluate

g_{1,2}'(2)

. (d) Discuss how varying the parameter

b

(with

a

fixed at

1

) affects the magnitude of

g_{1,b}'(2)

. Provide reasoning based on your formula from part (b).