If $f$ is twice differentiable function and $|f|<\alpha, |f''|<\beta$ in the domain $x>a$, then which of the fallowing may be upper bound for $|f'|$...
a) $\sqrt{\alpha\beta}$
b) $2.\sqrt{\alpha\beta}$
c) $\frac{1}{2}\sqrt{\alpha\beta}$
d) $4.\sqrt{\alpha\beta}$.
By taking $f(x)=\sin x$ c) is discarded. For other options, if a) is true then b),d) is also true. if b) is true then so d). So, the answer could be d).
I were trying to prove by Mean Value Theorem, but didn't get answer. Help is needed, Thankx in advance.