Let $f:\mathbb R \to \mathbb R$, with $f \in C^1$ and $f'>0$. Suppose there exists $x_0 \in \mathbb R$ such that $f(x_0)>0$.
Prove that $\space \space$ $\int_0^{+\infty} f(t)dt$ $\space$ is divergent.
I don't have a any idea how could I prove this.
I thought about series, and one necessary condition for a series to converge, which is $lim_{n \to +\infty} a_n=0$, maybe I could think of an analogue condition for this integral, i.e, $lim_{x \to +\infty} f(x)=0$. I am not so sure if what I am saying is correct, if I follow this path, I would not be using the hypothesis $f \in C^1$. I would appreciate any suggestions on how could I solve the problem.
Hint: If $f' > 0$, then $$y > x_0 \implies f(y) > f(x_0)$$ Do you see why, and how to use this fact?