The number of continuous functions $f:[0,1]\to\mathbb R$ that satisfy $\int_0^1xf(x)\,dx=\frac13+\frac14\int_0^1(f(x))^2\,dx$ 
90) The number of continuous functions $f:[0,1]\to\mathbb R$ that satisfy
  $$\int_0^1xf(x)\,dx=\frac13+\frac14\int_0^1(f(x))^2\,dx$$
  is
  A) 0
  B) 1
  C) 2
  D) $\infty$

How to approach this sum? I thought of using Newton-Leibniz but the limits are constants, so that approach failed.

Another way:
$$\int^{1}_{0}4xf(x)dx-\int^{1}_{0}(f(x))^2dx=\frac{4}{3}$$
$$\int^{1}_{0}f(x)\bigg(4x-f(x)\bigg)dx\leq \frac{1}{4}\int^{1}_{0}\bigg[f(x)+4x-f(x)\bigg]^2dx=\frac{4}{3}.$$
Equality hold when $f(x)=2x$
In $2$ line earlilier i have used the inequality $$ab\leq \frac{(a+b)^2}{4}$$ equality hold when $a=b.$
