e-x-gt-1-x-x-gt-0-set-x-pi-e-1-e-pi-e-1-gt-pi-e-e-pi-e-gt-pi-e-pi-e-e-gt-pi-e-e

Question Number 189886 by mnjuly1970 last updated on 23/Mar/23

e^{x} > 1 + x (\forall x > 0)

s e t : x = \sqrt{\frac{π}{e}} - 1

e^{\frac{\sqrt{π}}{\sqrt{e}} - 1} > \frac{\sqrt{π}}{\sqrt{e}} \Rightarrow e^{\frac{\sqrt{π}}{\sqrt{e}}} > \sqrt{π}

{(e^{\frac{\sqrt{π}}{\sqrt{e}}})}^{\sqrt{e}} > \sqrt{π}^{\sqrt{e}} \Rightarrow e^{\sqrt{π}} > \sqrt{π}^{\sqrt{e}}