inequalities, monotonicity and functions

Question

I have the function: $$f(x)=e^{4x}-8x^2$$ Now I want to prove that $4f(2x)<3f(x)+f(5x),\forall\text{ }x\in(0,+\infty)$. I can easily prove that $f$ is strictly increasing at $\mathbb{R}$ and I did the following: $$4f(2x)<3f(x)+f(5x)\iff 3f(2x)+f(2x)<3f(x)+f(5x)\iff$$ $$3[f(2x)-f(x)]<f(5x)-f(2x)$$ My idea is to find another function $g$ in order to use the monotonicity. Any help?

Answer 1

Idea:

Calculate a derviative of

$$g(x)=3f(x)+f(5x)-4f(2x)$$

and show that it is $>0$ for each $x>0$. Since $g(0) = 0$ you are done.

Answer 2

Hint: Note for $x>0$, $\; f’’(x)=16(e^{4x}-1)>0$, so this follows directly from Jensen’s inequality.