STEP II 1997 question 7 solution

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I'm not going to call this a solution, as I don't know how to do sketches here.

But $y^2 = x^2(a^2-x^2) = \frac{a^4}{4} - (x^2-\frac{a^2}{2})^2$ and so the maximum value of is , attained when . This gives minimum, maximum values for y of $\pm a^2/2$ . Observe also y = 0 when x=0, x=a, and I think you've all you need for a sketch...

Non-solution by DFranklin

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