Example of function which is twice differentiable with $f,f''$ strictly increasing but $\lim_{x\to \infty}f(x)\neq \infty$

Question

I wanted to find Example of function which is twice differentiable with $f,f''$ strictly increasing but $\lim_{x\to \infty}f(x)\neq \infty$.

My usual notion fails for above statement . As I thought if $f$ is strictly increasing and $f''$ strictly incresing means $f'$ should also in increase.

Where is my mistake in my thinking ?

Any help will be appreciated

Answer 1

Isn't $-e^{-x}$ such an example?

Answer 2

What about the simpler

$$f(x)=-\frac1x$$

Answer 3

Your mistake in your thinking is to believe that "strictly increasing" means "positive".

Going from negative to less negative is increasing. So you can have $f''$ strictly increasing but negative, with a decreasing $f'$ (but positive), hence an increasing $f$ as well.

You need to choose a function that does all that (see Kavi example).