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Activity9.4Sign Tables for the First Derivative

Once you have determined the critical numbers of a function, the next thing you might want to determine is the behavior of the function at each of its critical numbers. One way you could do that involves a sign table for the first derivative of the function.

Subsection9.4.1Exercises

1

The first derivative of \(\fe{f}{x}=x^3-21x^2+135x-24\) is \(\fe{\fd{f}}{x}=3(x-5)(x-9)\).

The critical numbers of \(f\) are trivially shown to be \(5\) and \(9\). Copy Table 9.4.1 onto your paper and fill in the missing information. Then state the local minimum and maximum points on \(f\). Specifically address both minimum and maximum points even if one and/or the other does not exist. Remember that points on the plane are represented by ordered pairs. Make sure that you state points on \(f\) and not \(\fd{f}\)!

Interval Sign of \(\fd{f}\) Behavior of \(f\)
\(\ointerval{-\infty}{5}\) \(\phantom{\text{negative}}\) \(\phantom{\text{decreasing}}\)
\(\ointerval{5}{9}\)
\(\ointerval{9}{\infty}\)
Table9.4.1\(\fe{\fd{f}}{x}=3(x-5)(x-9)\)

The first derivative of the function \(\fe{g}{t}=\frac{\sqrt{t-4}}{(t-1)^2}\) is \begin{equation*}\fe{\fd{g}}{t}=\frac{-3(t-5)}{2(t-1)^3\sqrt{t-4}}\text{.}\end{equation*}

2

State the critical numbers of \(g\); you do not need to show a formal determination of the critical numbers. You do need to write a complete sentence.

3

Copy Table 9.4.2 onto your paper and fill in the missing information. Then state the local minimum and maximum points on \(k\).

Specifically address both minimum and maximum points even if one and/or the other does not exist.

Interval Sign of \(\fd{g}\) Behavior of \(g\)
\(\ointerval{4}{5}\)
\(\ointerval{5}{\infty}\)
Table9.4.2\(\fe{\fd{g}}{t}=\frac{-3(t-5)}{2(t-1)^3\sqrt{t-4}}\)
4

Why did we not include any part of the interval \(\ointerval{-\infty}{4}\) in Table 9.4.2?

5

Formally, we say that \(\fe{g}{t_0}\) is a local minimum value of \(g\) if there exists an open interval centered at \(t_0\) over which \(\fe{g}{t_0}\lt\fe{g}{t}\) for every value of \(t\) on that interval (other than \(t_0\), of course). Since \(g\) is not defined to the left of \(4\), it is impossible for this definition to be satisfied at \(4\); hence \(g\) does not have a local minimum value at \(4\). State the local minimum and maximum points on \(g\). Specifically address both minimum and maximum points even if one and/or the other does not exist.

6

Write a formal definition for a local maximum point on \(g\).

The first derivative of the function \(\fe{k}{x}=\frac{\sqrt[3]{(x-2)^2}}{x-1}\) is \begin{equation*}\fe{\fd{k}}{x}=\frac{4-x}{3(x-1)^2\sqrt[3]{x-2}}\text{.}\end{equation*}

7

State the critical numbers of \(k\); you do not need to show a formal determination of the critical numbers. You do need to write a complete sentence.

8

Copy Table 9.4.3 onto your paper and fill in the missing information.

Interval Sign of \(\fd{k}\) Behavior of \(k\)
\(\ointerval{-\infty}{1}\)
\(\ointerval{1}{2}\)
\(\ointerval{2}{4}\)
\(\ointerval{4}{\infty}\)
Table9.4.3\(\fe{\fd{k}}{t}=\frac{4-x}{3(x-1)^2\sqrt[3]{x-2}}\)
9

The number \(1\) was included as an endpoint in Table 9.4.3 even though \(1\) is not a critical number of \(k\). Why did we have to include the intervals \(\ointerval{-\infty}{1}\) and \(\ointerval{1}{2}\) in the table as opposed to just using the single interval \(\ointerval{-\infty}{2}\)?

Perform each of the following for the functions in Exercises 9.4.1.10–9.4.1.12.

  • Formally establish the critical numbers of the function.

  • Create a table similar to Tables 9.4.19.4.3. Number the tables. Don't forget to include table headings and column headings.

  • State the local minimum points and local maximum points on the function. Make sure that you explicitly address both types of points even if there are none of one type and/or the other.

10

\(\fe{k}{x}=x^3+9x^2-10\)

11

\(\fe{g}{t}=(t+2)^3(t-6)\)

12

\(\fe{F}{x}=\dfrac{x^2}{\fe{\ln}{x}}\)

Consider a function \(f\) whose first derivative is \(\fe{\fd{f}}{x}=(x-9)^4\).

13

Is \(9\) definitely a critical number of \(f\)? Explain why or why not.

14

Other than at \(9\), what is always the sign of \(\fe{\fd{f}}{x}\)? What does this sign tell you about the function \(f\)?

15

What type of point does \(f\) have at \(9\)? (Hint, draw a freehand sketch of the curve.)

16

How could the second derivative of \(f\) be used to confirm your conclusion in Exercise 9.4.1.15? Go ahead and do it.