Question 86
Let $f: [0, \pi/2] \to [0, \infty)$ be an increasing function. Prove that:
a. $$\int_0^{\pi/2} (f(x) - f(\pi/4))(\sin x - \cos x) dx \geq 0.$$
b. There exists $a \in [\pi/4, \pi/2]$ such that $\int_0^a f(x) \sin x dx = \int_0^a f(x) \cos x dx$.
a. $$\int_0^{\pi/2} (f(x) - f(\pi/4))(\sin x - \cos x) dx \geq 0.$$
b. There exists $a \in [\pi/4, \pi/2]$ such that $\int_0^a f(x) \sin x dx = \int_0^a f(x) \cos x dx$.