The answer is shown here by Rajada:
Proof for which exponent is greater
q1.) His solution seems nice but I can't understand how he gets to the second line from the first? (I understand the first line)
$$\frac{(e^a + e^b)}{2} \geq e^{\frac{(a+b)}{2}}\text{ (Using A.M.-G.M. inequality.)}$$ $$(e^a + e^b) > e^{\frac{(a+b+1)}{2}}\text{ (Using $4>e$).}$$
Can someone please explain.
Also his solution only applies to:
$$(e^a + e^b)> e^{(a+b+1)} \text{ when, }a+b+1 \leq 0.$$
q2.) Is it safe to assume that the direction of the inequality reverses when $\text{ when, }a+b+1 \geq 0.$
q1: Multiply the left-hand side by $2$ and the right-hand side by $e^{1/2}$. Since $2>e^{1/2}$ this preserves the inequality.
q2: No, because there's some slack in the preceding steps. For example the original inequality is always true when one of $a$ and $b$ is $0$, no matter what the other one is.