## The Second Derivative

Now that we've seen how the derivative and its zeroes can help us to locate local maxima and minima of functions, let's consider an example which will put things into a larger context.

An Example

Pictured below is the graph of the function and its derivative.

 Where are the local maxima and minima of the function? Where are the zeroes of the derivative? How does the behaviour of the derivative near a zero determine whether a point is a local maxima or minima of the function? Why is there one point which is a zero of the derivative yet not a local maxima or minima?

The Second Derivative
Let's consider another example before we introduce the second derivative. We have seen the function several times already and noted its local maximum and minimum value.

We detected the local maximum by noticing that for smaller values of , the derivative was positive, while for larger values of the derivative was negative. If we apply the language of increasing and decreasing functions to this situation and study the graph of the derivative, we see that the derivative is decreasing at the local maximum.

In the same way, at the local minimum, the derivative is negative for smaller values of and positive for larger values of . The graph shows us that the derivative is decreasing at this point.

If we remember that the derivative of a function tells us whether the function is increasing or decreasing, then we are now interested in the derivative of the derivative which we generally call the second derivative.

In the example above, we have that and . Now since the derivative is itself a function, we can differentiate it to obtain the second derivative which we denote as . Indeed, as we expect, we find that at the local maximum, . In other words, the second derivative of is negative and hence the derivative of is decreasing. This tells us that the critical point in question is a local maximum. Also, at the local minimum, we have . This means that the derivative is increasing at this critical point which is then a local minimum.