Let $f$ be a continuous strictly increasing function from $[0,\infty)$ onto $[0,\infty)$ and $g=f^{-1}$ (that is, $f(x)=y$ if and only if $g(y)=x$). Let $a,b>0$ and $a\ne b$. Then $$ \int_0^a f(x) dx + \int_0^b g(y) dy $$ is
(A) greater than or equal to $ab$, (B) less than $ab$, (C) always equal to $ab$,
(D) always equal to $\frac{af(a)+bg(b)}{2}$.
Though I am done with theory part I cannot think how to approach such problems. Please give me hints on how to start solving it. Thank you.
This visual presentation would suffice for a multiple-choice question. The condition holds for all $f$ that meets the condition in the problem statement. Try to draw the conclusion.