I want to find a function such that $f(x)$ is bounded by a constant $c$, i.e., $f(x)<c$. The derivative of the function must also be bounded by a constant, i.e., $f'(x)<c$ but the second derivative can be as large as you like. What is the method to find the answers to questions like these?
I know that $\sin (\theta ^2)$ has its first derivative as large as you like, is it possible to find some function whose derivative is $\sin (\theta ^2)$ so that it satisfies the above problem?
PS: If you have an answer, please tell how did you get the answer(the logic behind it)