Let be the cubic function s(x)=a x^ 3 +b x

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

Let

s

be the cubic function

s(x)=a x^{3}+b x,

where

a

and

b

are real constants. It is given that *

s(2)=8

, * the inverse function

s^{-1}

exists in a neighborhood of

x=2

, and

(s^{-1})'(8)=\dfrac1{24}

. Complete the tasks below, providing full justification. (1) Using

s(2)=8

, write an equation involving

a

and

b

. Explain the substitution used. (2) Determine

s'(x)

and, with the information

(s^{-1})'(8)=\dfrac1{24}

, establish a second equation in

a

and

b

. Carefully detail how the derivative of the inverse generates this equation. (3) Solve the system from parts (a) and (b) to find

a

and

b

. Show every algebraic step and check your solution. (4) Confirm that your values of

a

and

b

satisfy the original conditions, including the given value of

(s^{-1})'(8)

. State any conclusions about the local invertibility of

s

near

x=2

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