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Find all numbers $a$ such that $\forall x, a^x \ge 1+x$
I immediately thought of the inequality $e^x\ge 1+x$ and guessed that the answer was any number $a$ in $[e,\infty)$. After playing around with a graphing app I can see this is definitely true, although I can't explain why.
Why is $e$ specifically the smallest number such that the inequality holds? To my untrained eye, a quick scan of the equation does not give rise to anything involving the constant $e$. Maybe because $1+x$ is only a tangent to the equation $a^x$ when $a=e$? Thanks.