prove that $∃ c ∈ (0, 1)$ such that $c^2f(c) = 2\int_0^cxf(x) \,dx$ for a continuous function $f$
Prove that $\exists a, b\in(0,1)$ such that $\int_0^{a} xf(x)dx=0\text{ and }\int_0^bxf(x)dx=\frac{b^2f(b)}{2}.$