Question 47
Let $f:[0,\pi/2]\to[0,\infty)$ be increasing. Prove:
a) $\displaystyle\int_0^{\pi/2}(f(x)-f(\pi/4))(\sin x-\cos x)\,dx\ge0$.
b) There exists $a\in[\pi/4,\pi/2]$ such that $\displaystyle\int_0^a f(x)\sin x\,dx=\int_0^a f(x)\cos x\,dx$.
a) $\displaystyle\int_0^{\pi/2}(f(x)-f(\pi/4))(\sin x-\cos x)\,dx\ge0$.
b) There exists $a\in[\pi/4,\pi/2]$ such that $\displaystyle\int_0^a f(x)\sin x\,dx=\int_0^a f(x)\cos x\,dx$.