Extremum of a function

Question

I want to know whether it is possible that a point be an extremum without the derivative at that point being zero ?

I encountered this point when reading about the fact that entropy maximizes as function of energy but we know that $1/T=\frac{\partial S}{\partial E}$ (assuming Entropy is smooth and differentiable in the desired domain) which would make temperature infinite.

I already read a similar post on Physics Stackexchange but didn't get satisfactory answer.

Answer 1

It is possible, if your domain is compact with boundary or has a discontinuity: for the easiest example, think of a graph of a straight line y=x on $0 \leq x \leq 10$ having a maximum at (10, 10) and a minimum at (0,0).

If there's a discontinuity in the domain or the range (far more treacherous), an extremum could occur there (again, the graph of the function would 'break off' and continue elsewhere).

Answer 2

More ever of the points that derivative is equal to zero the the span boundaries also should be considered as critical points

Answer 3

Just pointing out another example, to add to all of this. The absolute value function does not have a definite value for the derivative at the Origin (the extremum), as the relevant limit does not exist at said point.

Answer 4

Consider $f(x) = |x|$. It has a minimum at $x= 0$, but the derivative isn't $0$ at $x=0$. Or consider the function $g:[-1,2]\to [0,4]$ via $g(x) = x^2$. It has a maximum at $x=2$ but $g'(2) = 4$.

These are exceptions of course. If $f$ is continuous, differentiable and $x$ is not an extreme point of a compact domain then an extremum and $x$ would imply that the derivative at $x$ is zero. That should be easy to show by definitions. (Intuitively if the derivative is positive or negative, the function is increasing or decreasing which means there will be points before and after which are larger on one side and smaller on the other.)