Real analysis : Problem related to inverse

Question

Let $f:[0,\infty)\rightarrow [0,\infty) $ be a continuous strictly increasing function and $g=f^{-1}$. Let $a,b>0$ and $a\neq b$. Then how $\displaystyle\int_{0}^af(x)dx+\int_0^bg(y)dy\geq ab.$ Any hint will be appreciated.

Answer 1

Description of the proof: Reflect the graph of $g$ with respect to the line $y=x$ and then observe that the area below the graph of $f$ and the $x$ axis (from $0$ to $a$) and the area between the reflected graph of $g$ and the $y$ axis (between $0$ and $b$) have a sum, which is larger than the area $ab$ of the square, that is covered by those areas.

A comment: This trick is standard and it is used to prove Young's inequality, which is usually applied to prove Holder's inequality.