Consider the rational function h(x)= m x 1+n x

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

Consider the rational function

h(x)=\frac{m x}{1+n x},

where

m

and

n

are positive real constants and

x\neq-\dfrac1n

. The following conditions hold: *

h(1)=3

. * The inverse function

h^{-1}

exists on an interval containing

x=1

, and

(h^{-1})'(3)=2

. Answer the prompts below. Show all calculus and algebraic justification. (1) Derive an equation involving

m

and

n

from the condition

h(1)=3

. Explain each step. (2) Compute

h'(x)

and, using

(h^{-1})'(3)=2

, obtain a second equation relating

m

and

n

. Clearly indicate how inverse and derivative information are connected. (3) Solve the system from parts (a) and (b) to determine the exact values of

m

and

n

. (4) Identify the largest open interval containing

x=1

on which

h

is one-to-one. Support your answer using appropriate calculus reasoning.