Activity 3.6.5

Consider the region in the $x$-$y$ plane that is bounded by the $x$-axis and the function $f(x) = 25-x^2$. Construct a rectangle whose base lies on the $x$-axis and is centered at the origin, and whose sides extend vertically until they intersect the curve $y = 25-x^2$. Which such rectangle has the maximum possible area? Which such rectangle has the greatest perimeter?

(Challenge: answer the same questions in terms of positive parameters $a$ and $b$ for the function $f(x) = b-ax^2$.)