We know if \(f\) is differentiable at \(x=a\) then the graph of \(f\) is locally linear at \(x=a\) and \(f'(a)\) is the slope of the tangent line at the point \((a, f(a))\). In this section, we explore how local behavior near a point can be extrapolated to global behavior over an interval. First, we review Definition incdeccnstdefn from Section ConstantandLinearFunctions:

Definition.

Let \(f\) be a function defined on an interval \(I\). Then \(f\) is said to be:

  • increasing on \(I\) if, whenever \(a < b\), then \(f(a) < f(b)\). (i.e., as inputs increase, outputs increase.)

    NOTE: The graph of an increasing function rises as one moves from left to right.

  • decreasing on \(I\) if, whenever \(a < b\), then \(f(a) > f(b)\). (i.e., as inputs increase, outputs decrease.)

    NOTE: The graph of a decreasing function falls as one moves from left to right.

  • constant on \(I\) if \(f(a) = f(b)\) for all \(a\), \(b\) in \(I\). (i.e., outputs don’t change with inputs.)

    NOTE: The graph of a function that is constant over an interval is a horizontal line.

Suppose a function satisfies \(f'(x) > 0\) for all \(x\) in an open interval \(I\). Then we know that not only is the graph of \(f\) locally linear on \(I\), but the slopes of all of the tangent lines are positive. This means that all of the tangent lines are increasing so it stands to reason that the function \(f\) is likewise increasing on \(I\). In other words, if a function is locally increasing on \(I\), then it is globally increasing on \(I\) as well.

We can apply the same reasoning above to situations where \(f'(x)<0\) for all \(x\) in \(I\), which implies \(f\) is decreasing on \(I\) or \(f'(x) = 0\) on \(I\), which implies \(f\) is constant on \(I\). In Calculus, you’ll learn this fact is a consequence of the Mean Value Theorem. In this text, we’ll just accept the following theorem is true and hope we’ve done enough hand-waving to deem it reasonable.

Thanks to GeoGebra, we can visualize Theorem firstderivatveandgraphs. In the first case below, we have two sections of a graph over which a function is increasing. By adjusting the slider, we can move the point across the graph, observe the corresponding tangent line, and see the value of the slope. Note that in both cases below, the slope of the tangent line, \(f'(x)\), is always positive.

\(f'(x) > 0\) for all \(x\) in \(I\).

Next, we have two sections of a graph over which a function is decreasing. Here, we observe the slopes of the tangent lines are always negative.

\(f'(x) < 0\) for all \(x\) in \(I\).

Last, but not least, we have the graph of a constant function. Here the slopes are always \(0\).

PIC

\(f'(x) = 0\) for all \(x\) in \(I\).

We can use Theorem firstderivatveandgraphs to help us determine the (open) intervals over which a function \(f\) is increasing, decreasing, and constant by making a sign diagram for the derivative \(f'\).

In order to avoid us having to go through the (somewhat lengthy) process of finding \(f'(x)\) using Definition derivativefcndefn, we’ll just use some properties of derivatives from Calculus behind the scenes and present you with both a function and its derivative. It’s time for an example.

We generalize our observations about local extrema in the following result.

1 Concavity and the Second Derivative

In section Section ??, we introduced the notion of concavity. In that section, we described curves as being concave up over an interval if it resembles a portion of a ‘\(\smile \)’ shape and concave down over an interval if resembles part of a ‘\(\frown \)’ shape. Now that we’ve had some exposure to Calculus, we can more precisely define these notions.

If we take the time to study a generic concave up curve, the ‘\(\smile \)’ shape can be divided into a decreasing and increasing arc. Using the GeoGebra interactive below, we notice that as we move the slider from left to right on the decreasing portion of the curve, the slopes are increasing towards \(0\).

Moving the slider on the second portion of the graph, we see the slopes are increasing from \(0\) as we move fron left to right.

In both of these cases, the slopes of the tangent line are increasing.

Likewise, we can dissect a generic ‘\(\frown \)’ shape curve into an increasing and decreasing arc. Here, as we move from left to right on the increasing portion, the slopes are decreasing to \(0\),

Moving the slider on the second portion of the graph from left to right, we find the slopes continuing to decrease from \(0\).

Here, the slopes of the tangent line are decreasing.

We know from Theorem 1 that the derivative of a function can tell us where that function is increasing and decreasing. Since the function which gives us the slopes of tangent lines is the derivative, \(f'(x)\), we could use the derivative of \(f'(x)\) to determine where the slopes of the tangent lines were increasing and decreasing. This leads us to define the second derivative, \(f''(x)\) as the derivative of \(f'(x)\).

We present the following theorem without proof, but hopefully sufficiently motivated.

Note that we can use concavity to help us distinguish local extrema.

For the function above, both \(f'(-1) = 0\) and \(f'(3) = 0\). Note that \(f''(-1) < 0\) which means \(f\) is concave down there. This forces \(f\) to have a local maximum at \((-1,6)\). Likewise, \(f''(3) > 0\) which means \(f\) is concave up there. This forces \(f\) to have a local minimum at \((3,-22)\). We generalize this observation below.

Our last example offers a twist on these sorts of curve-sketching problems.

2 Exercises