Let f be a twice differentiable function on R such that both $f'$ and $f''$ are strictly positive on R. Then $\lim_{x\to \infty} f(x)= \infty$.
I know the result is false and i can think of some example in my mind...but I can't get a concrete counterexample.
If this result is true please exaplain why? I hope it won't be
Since $f'$ is strictly positive, and increasing, then we have $f'(x)>k=f'(0)$ for all positive $x$. That means that, for positive $x$, $f(x)\ge g(x)=f(0)+kx$. We know that $g\to\infty$ as $x$ grows, so that's a proof.