Is there inequality of the type $e^x+e^y \le e^{x+y}$

Question

Is there inequality of the type $e^x+e^y \le e^{x+y}$ or $e^x+e^y \ge e^{x+y}$

You should look into convex functions. – Cameron Williams Sep 25 '14 at 00:28 — Cameron Williams, Sep 25 '14 at 00:28
Consider $x\le y\le 0$. What happens? – abiessu Sep 25 '14 at 00:30 — abiessu, Sep 25 '14 at 00:30

score 4 · Accepted Answer · answered Sep 25 '14 at 00:29

4

Because $e^x$ is convex, you have $$e^{\frac{x+y}{2}}\le \frac{e^x+e^y}{2}$$ which can be rearranged as $$2\sqrt{e^{x+y}}\le e^x+e^y$$

answered Sep 25 '14 at 00:29

vadim123

1 Answers1