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 \(a^{x-\varepsilon}< 1+x-\varepsilon\).

Difficulty level: Easy task (using definitions and simple reasoning)
Routine calculation training
Cs translation
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