The function f(x), defined for x>=0 has the following property: For any real number a>=0, the area between the graph of y=f(x) for 0<=x<=a and the graph of y=sqrt(x) 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) = 2sqrt(x) + kx for some constant k.

Guest Jul 14, 2021