Maximizing the area of a rectangle inscribed in an isosceles right triangle

As you drag the [b][color=#000000]black point[/color][/b] along the hypotenuse of this isosceles right triangle, the rectangle's area changes. What value of [math]x[/math] maximizes the area of the rectangle? Observe that the triangle has vertices at (-1,0), (0,1), and (1,0).
[i]Developed for use with Thomas' Calculus, published by Pearson.[/i]
– Tim Brzezinski

Information: Maximizing the area of a rectangle inscribed in an isosceles right triangle