Which number is bigger: $e^{\sqrt{5}} + e^{\sqrt{7}}$ or $2e^{\sqrt{6}}$? Let $f:(0, + \infty) \rightarrow \mathbb{R}$ be $f(x) = e^{\sqrt{x}}$.
Which number is bigger:  $e^{\sqrt{5}} + e^{\sqrt{7}}$ or $2e^{\sqrt{6}}$?
I would like to ask for a hint on how to approach this question, I don't have an idea.

Let $f(x)=e^{\sqrt{x}}$, so the second derivative is:
$$f''(x)=\frac{e^{\sqrt{x}}}{4x^{3/2}}(\sqrt{x}-1)$$
Since $f''(x)>0$ for $x\in (1,\infty)$, it is concave up (convex). So we have
$$\frac{f(x)+f(y)}{2}>f\left(\frac{x+y}{2}\right)$$
Let $x=5, y=7$
$$\frac{f(5)+f(7)}{2}>f\left(6\right)$$
