Active Calculus

Subsection 3.1.1 Critical numbers and the first derivative test

If a continuous function has a relative maximum at \(c\text{,}\) then it is both necessary and sufficient that the function change from being increasing just before \(c\) to decreasing just after \(c\text{.}\) A continuous function has a relative minimum at \(c\) if and only if the function changes from decreasing to increasing at \(c\text{.}\) (See Figure 3.1.5.) There are only two possible ways for these changes in behavior to occur: either \(f'(c) = 0\) or \(f'(c)\) is undefined. Because these values of \(c\) are so important, we call them critical numbers.

Critical numbers are the only possible locations where the function \(f\) may have relative extremes. Note that not every critical number produces a maximum or minimum; in the middle graph of Figure 3.1.5, the function pictured there has a horizontal tangent line at the noted point, but the function is increasing before and increasing after, so the critical number does not yield a maximum or minimum.

When \(c\) is a critical number, we say that \((c,f(c))\) is a critical point of the function, or that \(f(c)\) is a critical value . The first derivative test summarizes how sign changes in the first derivative (which can only occur at critical numbers) indicate the presence of a local maximum or minimum for a given function.

Example 3.1.7.

Let \(f\) be a function whose derivative is given by the formula \(f'(x) = e^{-2x}(3-x)(x+1)^2\text{.}\) Determine all critical numbers of \(f\) and decide whether a relative maximum, relative minimum, or neither occurs at each.

Solution.

Since we already have \(f'(x)\) written in factored form, it is straightforward to find the critical numbers of \(f\text{.}\) Because \(f'(x)\) is defined for all values of \(x\text{,}\) we need only determine where \(f'(x) = 0\text{.}\) From the equation

\begin{equation*} e^{-2x}(3-x)(x+1)^2 = 0 \end{equation*}

and the zero product property, it follows that \(x = 3\) and \(x = -1\) are critical numbers of \(f\text{.}\) (There is no value of \(x\) that makes \(e^{-2x} = 0\text{.}\))

Next, to apply the first derivative test, we'd like to know the sign of \(f'(x)\) at inputs near the critical numbers. Because the critical numbers are the only locations at which \(f'\) can change sign, it follows that the sign of the derivative is the same on each of the intervals created by the critical numbers: for instance, the sign of \(f'\) must be the same for every \(x \lt -1\text{.}\) We create a first derivative sign chart to summarize the sign of \(f'\) on the relevant intervals, along with the corresponding behavior of \(f\text{.}\)

To produce the first derivative sign chart in Figure 3.1.8 we identify the sign of each factor of \(f'(x)\) at one selected point in each interval. For instance, for \(x \lt -1\text{,}\) we could determine the sign of \(e^{-2x}\text{,}\) \((3-x)\text{,}\) and \((x+1)^2\) at the value \(x = -2\text{.}\) We note that both \(e^{-2x}\) and \((x+1)^2\) are positive regardless of the value of \(x\text{,}\) while \((3-x)\) is also positive at \(x = -2\text{.}\) Hence, each of the three terms in \(f'\) is positive, which we indicate by writing “\(+++\text{.}\)” Taking the product of three positive terms results in a positive value for \(f'\text{,}\) which we denote by the “\(+\)” in the interval to the left of \(x = -1\text{.}\) And, since \(f'\) is positive on that interval, we know that \(f\) is increasing, so we write “INC” to represent the behavior of \(f\text{.}\) In a similar way, we find that \(f'\) is positive and \(f\) is increasing on \(-1 \lt x \lt 3\text{,}\) and \(f'\) is negative and \(f\) is decreasing for \(x \gt 3\text{.}\)

Now we look for critical numbers at which \(f'\) changes sign. In this example, \(f'\) changes sign only at \(x = 3\text{,}\) from positive to negative, so \(f\) has a relative maximum at \(x = 3\text{.}\) Although \(f\) has a critical number at \(x = -1\text{,}\) since \(f\) is increasing both before and after \(x = -1\text{,}\) \(f\) has neither a minimum nor a maximum at \(x = -1\text{.}\)

Subsection 3.1.2 The second derivative test

Recall that the second derivative of a function tells us several important things about the behavior of the function itself. For instance, if \(f''\) is positive on an interval, then we know that \(f'\) is increasing on that interval and, consequently, that \(f\) is concave up, so throughout that interval the tangent line to \(y = f(x)\) lies below the curve at every point. At a point where \(f'(p) = 0\text{,}\) the sign of the second derivative determines whether \(f\) has a local minimum or local maximum at the critical number \(p\text{.}\)

In Figure 3.1.9, we see the four possibilities for a function \(f\) that has a critical number \(p\) at which \(f'(p) = 0\text{,}\) provided \(f''(p)\) is not zero on an interval including \(p\) (except possibly at \(p\)). On either side of the critical number, \(f''\) can be either positive or negative, and hence \(f\) can be either concave up or concave down. In the first two graphs, \(f\) does not change concavity at \(p\text{,}\) and in those situations, \(f\) has either a local minimum or local maximum. In particular, if \(f'(p) = 0\) and \(f''(p) \lt 0\text{,}\) then \(f\) is concave down at \(p\) with a horizontal tangent line, so \(f\) has a local maximum there. This fact, along with the corresponding statement for when \(f''(p)\) is positive, is the substance of the second derivative test.

In the event that \(f''(p) = 0\text{,}\) the second derivative test is inconclusive. That is, the test doesn't provide us any information. This is because if \(f''(p) = 0\text{,}\) it is possible that \(f\) has a local minimum, local maximum, or neither.²

Just as a first derivative sign chart reveals all of the increasing and decreasing behavior of a function, we can construct a second derivative sign chart that demonstrates all of the important information involving concavity.

Example 3.1.10.

Let \(f(x)\) be a function whose first derivative is \(f'(x) = 3x^4 - 9x^2\text{.}\) Construct both first and second derivative sign charts for \(f\text{,}\) fully discuss where \(f\) is increasing and decreasing and concave up and concave down, identify all relative extreme values, and sketch a possible graph of \(f\text{.}\)

Solution.

Since we know \(f'(x) = 3x^4 - 9x^2\text{,}\) we can find the critical numbers of \(f\) by solving \(3x^4 - 9x^2 = 0\text{.}\) Factoring, we observe that

\begin{equation*} 0 = 3x^2(x^2 - 3) = 3x^2(x+\sqrt{3})(x-\sqrt{3})\text{,} \end{equation*}

so that \(x = 0, \pm\sqrt{3}\) are the three critical numbers of \(f\text{.}\) The first derivative sign chart for \(f\) is given in Figure 3.1.11.

We see that \(f\) is increasing on the intervals \((-\infty, -\sqrt{3})\) and \((\sqrt{3}, \infty)\text{,}\) and \(f\) is decreasing on \((-\sqrt{3},0)\) and \((0, \sqrt{3})\text{.}\) By the first derivative test, this information tells us that \(f\) has a local maximum at \(x = -\sqrt{3}\) and a local minimum at \(x = \sqrt{3}\text{.}\) Although \(f\) also has a critical number at \(x = 0\text{,}\) neither a maximum nor minimum occurs there since \(f'\) does not change sign at \(x = 0\text{.}\)

Next, we move on to investigate concavity. Differentiating \(f'(x) = 3x^4 - 9x^2\text{,}\) we see that \(f''(x) = 12x^3 - 18x\text{.}\) Since we are interested in knowing the intervals on which \(f''\) is positive and negative, we first find where \(f''(x) = 0\text{.}\) Observe that

\begin{equation*} 0 = 12x^3 - 18x = 12x\left(x^2 - \frac{3}{2}\right) = 12x\left(x+\sqrt{\frac{3}{2}}\right)\left(x-\sqrt{\frac{3}{2}}\right)\text{.} \end{equation*}

This equation has solutions \(x = 0, \pm\sqrt{\frac{3}{2}}\text{.}\) Building a sign chart for \(f''\) in the exact same way we do for \(f'\text{,}\) we see the result shown in Figure 3.1.12.

Therefore, \(f\) is concave down on the intervals \((-\infty, -\sqrt{\frac{3}{2}})\) and \((0, \sqrt{\frac{3}{2}})\text{,}\) and concave up on \((-\sqrt{\frac{3}{2}},0)\) and \((\sqrt{\frac{3}{2}}, \infty)\text{.}\)

Putting all of this information together, we now see a complete and accurate possible graph of \(f\) in Figure 3.1.13.

The point \(A = (-\sqrt{3}, f(-\sqrt{3}))\) is a local maximum, because \(f\) is increasing prior to \(A\) and decreasing after; similarly, the point \(E = (\sqrt{3}, f(\sqrt{3})\) is a local minimum. Note, too, that \(f\) is concave down at \(A\) and concave up at \(B\text{,}\) which is consistent both with our second derivative sign chart and the second derivative test. At points \(B\) and \(D\text{,}\) concavity changes, as we saw in the results of the second derivative sign chart in Figure 3.1.12. Finally, at point \(C\text{,}\) \(f\) has a critical point with a horizontal tangent line, but neither a maximum nor a minimum occurs there, since \(f\) is decreasing both before and after \(C\text{.}\) It is also the case that concavity changes at \(C\text{.}\)

While we completely understand where \(f\) is increasing and decreasing, where \(f\) is concave up and concave down, and where \(f\) has relative extremes, we do not know any specific information about the \(y\)-coordinates of points on the curve. For instance, while we know that \(f\) has a local maximum at \(x = -\sqrt{3}\text{,}\) we don't know the value of that maximum because we do not know \(f(-\sqrt{3})\text{.}\) Any vertical translation of our sketch of \(f\) in Figure 3.1.13 would satisfy the given criteria for \(f\text{.}\)

Points \(B\text{,}\) \(C\text{,}\) and \(D\) in Figure 3.1.13 are locations at which the concavity of \(f\) changes. We give a special name to any such point.

Just as we look for locations where \(f\) changes from increasing to decreasing at points where \(f'(p) = 0\) or \(f'(p)\) is undefined, so too we find where \(f''(p) = 0\) or \(f''(p)\) is undefined to see if there are points of inflection at these locations.

At this point in our study, it is important to remind ourselves of the big picture that derivatives help to paint: the sign of the first derivative \(f'\) tells us whether the function \(f\) is increasing or decreasing, while the sign of the second derivative \(f''\) tells us how the function \(f\) is increasing or decreasing.

Activity 3.1.3.

Suppose that \(g\) is a function whose second derivative, \(g''\text{,}\) is given by the graph in Figure 3.1.15.

1. Find the \(x\)-coordinates of all points of inflection of \(g\text{.}\)

2. Fully describe the concavity of \(g\) by making an appropriate sign chart.

3. Suppose you are given that \(g'(-1.67857351) = 0\text{.}\) Is there is a local maximum, local minimum, or neither (for the function \(g\)) at this critical number of \(g\text{,}\) or is it impossible to say? Why?

4. Assuming that \(g''(x)\) is a polynomial (and that all important behavior of \(g''\) is seen in the graph above), what degree polynomial do you think \(g(x)\) is? Why?

Hint.

1. What must be true of \(g''(x)\) at a point of inflection?

2. Use the given graph to decide where \(g''\) is positive and negative.

3. What does the second derivative test say?

4. Can you guess a formula for \(g''(x)\) based on its graph?

As we will see in more detail in the following section, derivatives also help us to understand families of functions that differ only by changing one or more parameters. For instance, we might be interested in understanding the behavior of all functions of the form \(f(x) = a(x-h)^2 + k\) where \(a\text{,}\) \(h\text{,}\) and \(k\) are parameters. Each parameter has considerable impact on how the graph appears.

Subsection 3.1.3 Summary

• The critical numbers of a continuous function \(f\) are the values of \(p\) for which \(f'(p) = 0\) or \(f'(p)\) does not exist. These values are important because they identify horizontal tangent lines or corner points on the graph, which are the only possible locations at which a local maximum or local minimum can occur.

• Given a differentiable function \(f\text{,}\) whenever \(f'\) is positive, \(f\) is increasing; whenever \(f'\) is negative, \(f\) is decreasing. The first derivative test tells us that at any point where \(f\) changes from increasing to decreasing, \(f\) has a local maximum, while conversely at any point where \(f\) changes from decreasing to increasing \(f\) has a local minimum.

• Given a twice differentiable function \(f\text{,}\) if we have a horizontal tangent line at \(x = p\) and \(f''(p)\) is nonzero, the sign of \(f''\) tells us the concavity of \(f\)and hence whether \(f\) has a maximum or minimum at \(x = p\text{.}\) In particular, if \(f'(p) = 0\) and \(f''(p) \lt 0\text{,}\) then \(f\) is concave down at \(p\) and \(f\) has a local maximum there, while if \(f'(p) = 0\) and \(f''(p) \gt 0\text{,}\) then \(f\) has a local minimum at \(p\text{.}\) If \(f'(p) = 0\) and \(f''(p) = 0\text{,}\) then the second derivative does not tell us whether \(f\) has a local extreme at \(p\) or not.