Implicitly

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

$X(r)$ is defined implicitly by the quadratic relationship
$$
X^2r^2-Xr-r+1=0
$$
Part 1: Which of the choices $r=1,-1,100$ give real values for $X(r)$?

Part 2: What is the range of values of $r$ for which $X(r)$ takes real values?
What happens when $r=0$?

Part 3: Sketch the overall shape of $X(r)$ against $r$ and find the maximum and minimum values of $X(r)$.

Note: You could numerically find a sensible conjecture for the minimum and maximum values of $X(r)$, but to prove this you will need to use calculus.


The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
Copyright © 1997 - 2012. University of Cambridge. All rights reserved.
NRICH is part of the family of activities in the Millennium Mathematics Project.