$\int_a^bf(x) dx=\int_a^bf^\prime(x) dx$

Question

I am facing problem in the following exercise. I don't know how to tackle this problem. Can anyone please help me?

The only thing I can say $$ \int_a^bf(x) dx=\int_a^bf^\prime(x) dx \implies \int_a^bf(x) dx=f(b)-f(a). $$

Answer 1

There is only one function that fits this case (the questions implies D is true, so I'll run with it) which is that $f(x)=e^x$. Therefore we must determine if the image of $f(x)$ (which is $\mathbb{R}^+$) is compact. This leads in to a topological argument and that was a class that didn't go so well for me.