Question 1: Let $f(x)$ be a continuous function over $[0, \pi]$. Then the definite integration of $f(x)\sin{x}$ and $f(x)\cos{x}$ over $[0, \pi]$ are all 0.
Prove that $f(x)$ has at least two zeros over $(0,\pi)$.
It is easy to see that $f(x)$ one root over $(0, \pi)$ But it is difficult to show $f(x)$ has at least two roots over $(0, \pi)$.
Question 2:
Let $f(x)$ be a continuous function over $[a, b]$. Then the definite integration of $f(x)$ and $f(x)g(x)$ over $[a, b]$ are all 0. And g'(x)~=0(over $(a, b)$).
Prove that $f(x)$ has at least two zeros over $(a,b)$.
