Integral of a function and its inverse - Real analysis

The following questing concerns a strictly increasing function f on [0,a] st f(0)=0



I think I've done the first part, the second I could do with some help please. I'm pretty sure there's a typo in the hint (first inequality on the final line).Thanks

I guess they're inviting you to consider the integrals $\int_0^\alpha + \int_0^{f(\alpha)} + \int_{f(\alpha)}^\beta$ if I've read the inequalities properly.

I agree, but I still can't work out how to use the hint to get the required inequality

Doesn't it follow trivially from part (a) + the inequality that they suggest?

But I need to show the inequality they suggest? That's really what I don't understand how to do.

That is, I know how to complete the question once I've established the hint inequality holds.