# what is the relationship between he second derivative of a function and its stationary points.  Can you explain this to me?

tiburtius | Certified Educator

Stationary points of differentiable function `f` are points `x` such that `f'(x)=0.`

Stationary points are often used when searching for local extrema (minimum or maximum) of a function. They are potential extrema of a function, however there are stationary points which are neither maximum nor minimum one such example is

`f(x)=x^3`

`f'(x)=3x^2`

`f'(0)=0` but zero is not an extremum which you can see on the graph below.

So to determine weather a stationary point is minimum or maximum we can use second derivative of a function. The next theorem explains how to do that.

Theorem

Let `c` be stationary point of twice differentiable function `f.` If `f''(c)<0,` then `f` has local maximum in `c.` If `f''(c)>0,` then `f` has local minimum in `c.`

This is because `f''(c)<0` means that function is decreasing at point `c` and since we know that `f'(c)=0` it follows function is increasing before `c` and decreasing after `c` implying that the point `c` is maximum.

You can also look at it from the point of convex and concave parts of a function which is also tied to the second derivative.

For more on this see the link below.