A question regarding fast-growing differentiable functions

Question

Suppose $f$ is a differentiable function from $\mathbb{R}$ to $\mathbb{R}$, such that the derivative $f'$ of $f$ grows towards infinity faster than $f$ itself. Also, suppose $g$ is a differentiable function such that $g \geq f$, in other words, $g$ is greater than or equal to $f$ pointwise. Does it then follow that the derivative $g'$ of $g$ grows faster than $g$ itself? Or is there a counterexample?

Edit: Also, what if we require that $g$ be not just greater than or equal to $f$, but strictly increasing?

Answer 1

Take $f=e^{2x}$, $g=e^{2x}(2+\sin(2x))$ as counterexample.

• $f'$ is faster than $f$, $\lim_{x\to\infty}\frac{f'}f>0$
• $g \ge f$
• $f,g$ strictly increasing
• $g'$ has no inflections, $g'=e^{2x}(4+2\sin(2x)+2\cos(2x)) > 0 $
• $g'$ is not faster than $g$, $\lim_{x\to\infty}\frac{g'}g=2+2\frac{\cos(2x)}{2+\sin(2x)}$ is undefined