What if the derivative of a function is positive for some range and negative for some other range?

Question

I have a function $f(x)$ whose derivative is positive for some range ($x<x_1$) while it becomes negative for $x>x_1$. Can I say that the function is concave and the maximum is achieved at $x=x_1$? Or there is some mistake in this reasoning?

Answer 1

Can I say that the function is concave?

No. You can say it is quasiconcave. For example the PDF of the normal distribution is not concave but it has positive derivative for $x < x_1$ and negative derivative for $x > x_1$ (where $x_1$ is the mean).

Is the maximum achieved at $x = x_1$?

Yes as was answered here.

Answer 2

If $f(x)$ is continuous and differentiable when $x\neq x_1$, and if $f'$ has opposite signs on each side of $x_1$, then $x=x_1$ does give the global maximum of $f$.

But $f$ does not have to be concave. It does not even have to be locally concave, in some small neighborhood around $x=x_1$.

Consider, for example,

$$f(x)=-\sqrt{|x|}$$

at $x=0$.

However, if $f$ is also differentiable at $x_1$, then $f'(x_1)$ must be zero. Now it must be the case that $f$ is locally concave, but $f$ does not have to be globally concave, as the example of the PDF of the normal distribution illustrates.