Inverse function — Derivative numerator analysis

Question

Show that has the same sign as for , and deduce the values of for which on .

This practice question was inspired by Question 14(b) in the 2024 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Set up the solution: replace the function value by y, swap x and y, then solve for y.Combine derivatives over common denominator .
Finish the calculation, then check that the result meets the question’s conditions.Both coefficients non-negative iff .
State the final answer clearly in the form the question requested.Answer: on precisely when .

Common mistake: changing so many features that the new example no longer tests the same mathematical idea.

Verification: At , numerator is ; at , numerator is .

Any mark labels are a TestMum study aid, not an official NESA marking allocation.