Estimation of the exponential function around zero
Task number: 3851
Let \( a \) be a positive real number for which the inequality \( 1 + x \leq a^x \) holds for all \( x \in \mathbb R \). Prove that then \( a = e \).
Solution
The function \( a^x \) is convex with the derivative \( a^x \ln a \), so at the point \( x = 0 \) it has the derivative \( \ln a \). Around zero, the function \( a^x \) can be approximated by the function \( 1 + x \ln a\).
When \( a<e \), then for sufficiently small \(\varepsilon\) it holds that \(a^{x+\varepsilon}< 1+x+\varepsilon\).
When \(a>e\), then for sufficiently small \(\varepsilon\) it holds that \(a^{x-\varepsilon}< 1+x-\varepsilon\).