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1997 Paper 2 Q7
D: 1600.0
B: 1516.0
curve sketching
implicit differentiation
optimisation
quartic
symmetry
stationary points
lemniscate-like curve
Let $$y^2=x^2(a^2-x^2),$$ where \(a\) is a real constant. Find, in terms of \(a\), the maximum and minimum values of \(y\). Sketch carefully on the same axes the graphs of \(y\) in the cases \(a=1\) and \(a=2\).
Solution: \begin{align*} && y^2 &= x^2a^2-x^2 \\ &&&= \frac{a^4}{4} -\left ( x^2 -\frac{a^2}{2} \right)^2 \end{align*} Therefore the maximum and minimum values of \(y\) are \(\pm \frac{a^2}2\)
