Age 16 to 18 Challenge Level:
$X(r)$ is defined implicitly by the quadratic relationship
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.