Finding $x$ where inequality is satisfied for sure

Question

I have following inequality $$ \frac{e^x}{x} \geq M$$ where $x$ is positive and the constant number $M$ is also positive. $e$ is Euler's number.

Does someone know some reference or have knowledge about finding $x$, where the inequality is fulfilled?

I tried to use Lambert W function, but it only leads to complex numbers, which is not helpful for my problem, since I need $x$ to be real and positive.

Thank You!

Answer 1

Hints:

• The minimum value of $\frac{e^x}{x}$ is $e$. For $M \le e$, all $x \ge 0$ satisfy the inequality.

• When $M > e$, then $e^x=Mx$ has two real solutions as found by Lambert $W$ function.

Answer 2

If $M\leq1$ then $e^x\geq1+x$ shows that the inequality holds for all $x$. If $M>1$ you can use $e^x\geq1+x+x^2/2$ to find the sufficient condition $$1-(M-1)x+x^2/2\geq0.$$ This holds for all $x$ if $(M-1)^2-2<0$ and it holds for all $$x>M-1+\sqrt{(M-1)^2-2}$$ otherwise.