Problem understanding proof that $e^a+e^b \geq e^{a+b}$

Question

The answer is shown here by Rajada:

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 $\text{ when, }a+b+1 \geq 0.$

SInce $4>e$, therefore, $2>e^{1/2}$ and so $(e^a+e^b)>\exp{[(a+b)/2+1/2]}$ — Kartik, Oct 25 '15 at 15:40
$e^a+e^b\ge 2e^{\frac{a+b}{2}}=4^{1/2}e^{\frac{a+b}{2}}>e^{1/2}e^{\frac{a+b}{2}}=e^{\frac{a+b+1}{2}}$. — user236182, Oct 25 '15 at 15:42

score 2 · Accepted Answer · answered Oct 25 '15 at 15:42

2

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.

answered Oct 25 '15 at 15:42

hmakholm left over Monica

286,031

OK Q1 clearer now but why does the inequality become strict? – Bazman Oct 25 '15 at 15:44
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@Bazman: Because $2$ is strictly larger than $e^{1/2}$. – hmakholm left over Monica Oct 25 '15 at 15:44
for Q2 Is there a more general solution that you can point me too? – Bazman Oct 25 '15 at 15:44
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@Bazman: I doubt there is any exact condition that is appreciably nicer than the inequality itself. – hmakholm left over Monica Oct 25 '15 at 15:45
You are probably correct but I am asked to prove the statement shown in the first line is it acceptable to state that this is the case because of the AM-GM theorem? – Bazman Oct 25 '15 at 15:49
@Bazman: On further thought, it is possible to simplify slightly. I have added an answer to the original question. – hmakholm left over Monica Oct 25 '15 at 16:09

Problem understanding proof that $e^a+e^b \geq e^{a+b}$

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