Given that $f=C[0,\pi]$, $$\int_{0}^{\pi} f(\theta)\cos\theta \,d\theta=\int_{0}^{\pi} f(\theta)\sin\theta \,d\theta=0 $$ prove that $f$ has at least two zero points at the interval$ (0,\pi)$.
This problem has already been asked by others,but there is only by contradiction. Can it be solved in a direct way?
if $\int_{0}^{x} f(x)\, dx$ has at least two extreme values then its derivative has at least two zero points naturally. But I can't prove that. I also thought that if the $\int_{0}^{x} f(x) \, dx$ has two pairs of equal values then its derivative has two zero points. So how to prove it? Thank you.
