At Great Value Expression

Question

Hi,

How can I find the greatest value of $2k\sqrt{(r-k)(r+k))}$ with parameter $r$ such that $(r>k)$?

Answer 1

Hint: Let us equivalently maximize the square of your expression. The square of your expression is equal to $$r^4-(r^2-2k^2)^2$$ (we "completed the square").