The function $f(x),$ defined for $x \ge 0,$ has the following properties:

$f(x) \ge \sqrt{x}$ for all $x \ge 0.$

The function $f(x)$ is increasing.

The area between the graph of $y = f(x)$ for $0 \le x \le a$ and the graph of $y = x^2$ is equal to the area between the same part of the graph and the $y$-axis. (In other words, the red area is equal to the blue area.)

(a) Find a differential equation that the function $f(x)$ satisfies. (In particular, this equation will involve $f(x)$ and $f'(x).$)

(b) Prove that $f(x) = 2x^2 + kx$ for some constant $k.$

maximum Aug 15, 2023