Maxima and Minima
Sketching Graphs
UBC Calculus Online Course Notes

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 $  f(x) = x^5 -x^4 - 
2x^3 $ and its derivative.


For your consideration:

  1. Where are the local maxima and minima of the function?

  2. Where are the zeroes of the derivative?

  3. How does the behaviour of the derivative near a zero determine whether a point is a local maxima or minima of the function?

  4. 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 $  f(x) = x^3 - x  $ several times already and noted its local maximum and minimum value.

We detected the local maximum by noticing that for smaller values of $  x  $ , the derivative was positive, while for larger values of $  x  $ 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 $  x  $ and positive for larger values of $  x  $ . 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 $ 
f(x) = x^3 - x  $ and $  f^\prime(x) = 3x^2 - 1  $ . Now since the derivative is itself a function, we can differentiate it to obtain the second derivative which we denote as $ 
f^{\prime\prime}(x) = 6x  $ . Indeed, as we expect, we find that at the local maximum, $  f^{\prime\prime}(-\frac{1}{\sqrt{3}}) = 
-\frac{6}{\sqrt{3}} < 0  $ . In other words, the second derivative of $  f  $ is negative and hence the derivative of $  f 
 $ is decreasing. This tells us that the critical point in question is a local maximum. Also, at the local minimum, we have $  f^{\prime\prime}(\frac{1}{\sqrt{3}}) = 
\frac{6}{\sqrt{3}} > 0  $ . This means that the derivative is increasing at this critical point which is then a local minimum.


For your consideration:

In the example above, the derivative has a minimum value at $  x = 0  $ . How is this reflected in the graph of the original function? How is this reflected by the second derivative?


What does the second derivative tell us?

We were lead to think about the second derivative by studying the behaviour of a function near its critical points. We can summarize that discussion by stating:

  • If $  x_0  $ is a critical point and $ 
f^{\prime\prime}(x_0) < 0  $ , then is a local maximum.
  • If $  x_0  $ is a critical point and $ 
f^{\prime\prime}(x_0) > 0  $ , then is a local minimum.

We can understand this more fully by thinking geometrically about the behaviour of the tangent lines. Let suppose that $ 
f^{\prime\prime} > 0  $ in some region. This means that the derivative $  f^\prime  $ , or the slope of the tangent lines, is increasing in this region. Here is a possible picture of the tangent lines.

If we now imagine the corresponding shape for the graph of the original function, it must be something like this:

We say that a graph which is curved in this way is concave up . The second derivative being positive implies that the graph is concave up since the slope of tangent lines must be increasing. Here is another example of a graph which is concave up.

Conversely, when the second derivative is negative, the slope of the tangent lines is decreasing. Here is a possible picture of the tangent lines:

If we now imagine the corresponding shape for the graph of the original function, it must be something like this:

We say that a graph is concave down when it is curved in this way. When the second derivative is negative, it implies that the graph is concave down since the slope of tangent lines must be decreasing. Here is another example of a graph which is concave down.


An important point on a graph is one which marks a transition between a region where the graph is concave up and one where it is concave down. We call such a point an inflection point . Have a look again at the graph of the function $  f(x) = x^3 - x 
 $ and see if you can find an inflection point. How does the behaviour of the tangent lines change at this point?


Summary

The second derivative tells us a lot about the qualitative behaviour of the graph.

  1. If the second derivative is positive at a point, the graph is concave up. If the second derivative is positive at a critical point, then the critical point is a local minimum.
  2. If the second derivative is negative at a point, the graph is concave down. If the second derivative is negative at a critical point, then the critical point is a local maximum.
  3. An inflection point marks the transition from concave up and concave down. The second derivative will be zero at an inflection point.