I want to prove the following inequality
Let $a$ and $b$ be positive real numbers and $f:\mathbb{R}\rightarrow\mathbb{R}$ a continuous function. Then,
$$\int_{0}^{a}f(x)dx+\int_{0}^{b}f^{-1}(y)dy\ge ab$$
I know that $(a-b)^2 \ge 0$ and so I easily get $2ab\le a^2 + b^2 $ and thus $ab\le a^2 + b^2 $. I feel like this should guide me to the answer, but I don't know how.
