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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.

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    [This is known as Young's inequality](https://en.wikipedia.org/wiki/Young's_inequality#Standard_version_for_increasing_functions). The [picture of the rectangle](https://i.stack.imgur.com/y5TKp.png) is the key idea. – 2017-02-16
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    More similar questions can be found with [Approach0](https://approach0.xyz/search/?q=%24%5Cint_%7B0%7D%5E%7Ba%7Df(x)dx%2B%5Cint_%7B0%7D%5E%7Bb%7Df%5E%7B-1%7D(y)dy%5Cge%20ab%24&p=1) – 2017-02-16

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