I'm stuck in proving the function $$f(x) = \vert x + e^x\vert $$ has a minimum.
This is what I did:
$$f'(x) = (1+e^x)\text{sgn}(1+e^x)$$
But this function is never zero, because the exponential is always positive. So when I study the sign of the derivative, it is always increasing.
Yet the plot of the funciton shows a "sort of" a minimum.
Adds
Since there is an absolute value I calculated the difference quotient for $f(x) > 0$ and $f(x) < 0$ obtaining the function is continuous when $x+e^x > 0$ and when $x+e^x < 0$
Yet I also understood @Martin R. comment but how to find that point where $f(x)$ is discontinuous?