Inqualitiy with exponent

Question

I was trying to prove this inequality $\exp(\frac{1}{\pi})+\exp(\frac{1}{e})\geq 2 \exp(\frac{1}{3})$ My attempt was using AM-GM mean $\exp(\frac{1}{\pi})+\exp(\frac{1}{e})\geq2 \exp(\frac{1}{2\pi e})$.

Answer 1

Since $$\pi+e<6,$$ by AM-GM and C-S and we obtain: $$e^{\frac{1}{\pi}}+e^{\frac{1}{e}}\geq2\sqrt{e^{\frac{1}{\pi}}\cdot e^{\frac{1}{e}}}=2\sqrt{e^{\frac{1}{\pi}+\frac{1}{e}}}\geq2\sqrt{e^{\frac{4}{\pi+e}}}>2\sqrt{e^{\frac{4}{6}}}=2e^{\frac{1}{3}}.$$