Learning Outcomes
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Determine where a differentiable function is increasing and decreasing and classify the critical points as local extrema.
Section 3.5 Derivative Tests (AD5)
Subsection 3.5.1 Activities
Definition 3.5.1.
We say that \(f(x)\) has a local maximum at \(x=c\) provided that \(f(c)\geq f(x)\) for all \(x\) near \(c\text{.}\) We also say that \(f(c)\) is a local maximum value for the function. On the other hand, we say that \(f(x)\) has a local minimum at \(x=c\) provided that \(f(c)\leq f(x)\) for all \(x\) near \(c\text{.}\) We also say that \(f(c)\) is a local minimum value for the function. The local maxima and minima are also known as the local extrema (or relative extrema) of the function.
Observation 3.5.2.
To find the extreme values of a function we can consider all its local extrema (local maxima and minima) and study them to find which one(s) give the largest and smallest values on the function. But how do you find the local/relative extrema? We will see that we can detect local extrema by computing the first derivative and finding the critical points of the function. By finding the critical points, we will produce a list of candidates for the extrema of the function.
Activity 3.5.3.
We have encountered several terms recently, so we should make sure that we understand how they are related. Which of the following statements are true?
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In a closed interval an endpoint is always a local extremum but it might or might not be a global extremum.
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In a closed interval an endpoint is always a global extremum.
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A critical point is always a local extremum but it might or might not be a global extremum.
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A local extremum only occurs where the first derivative is equal to zero.
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A local extremum always occurs at a critical point.
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A local extremum might occur at a critical point or at an endpoint of a closed interval.
Activity 3.5.4.
(a)
Sketch the graph of a continuous function that is increasing on \((-\infty, -2)\text{,}\) constant on the interval \((3,5)\text{,}\) and decreasing on the interval \((-2,3)\text{.}\)
(b)
How would you describe the derivative of the function on each interval?
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For \(x < -2\) we have \(f'(x) < 0\text{,}\) then \(f'(x) < 0\) on the interval \((-2,3)\text{,}\) and on the interval \((3,5)\) we have \(f'(x) > 0\text{.}\)
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For \(x < -2\) we have \(f'(x) > 0\text{,}\) then \(f'(x) < 0\) on the interval \((-2,3)\text{,}\) and on the interval \((3,5)\) we have \(f'(x) \) is undefined.
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For \(x < -2\) we have \(f'(x) > 0\text{,}\) then \(f'(x) < 0\) on the interval \((-2,3)\text{,}\) and on the interval \((3,5)\) we have \(f'(x)=0 \text{.}\)
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For \(x < -2\) we have \(f'(x) < 0\text{,}\) then \(f'(x) < 0\) on the interval \((-2,3)\text{,}\) and on the interval \((3,5)\) we have \(f'(x) \) is constant.
Activity 3.5.5.
Look back at the graph you made for Activity 3.5.4.
Which of the following best describes what is occurring when graph changes behavior?
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There is a critical point.
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There is a local maximum or minimum.
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The derivative is undefined.
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The derivative is equal to zero.
Observation 3.5.6.
Critical points detect changes in the behavior of a function. We will use critical points as "break points" in studying the behavior of a function. To understand what happens at the critical points we use the Derivative Tests.
Theorem 3.5.7. The First Derivative Test.
Suppose that \(x=c\) is a critical point of \(f(x)\) and that \(f(x)\) is continuous at \(x=c\text{.}\)
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If \(f'(x) > 0\) before \(x=c\) and \(f'(x) < 0\) after \(x=c\text{,}\) then a local maximum occurs at \(x=c\text{.}\)
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If \(f'(x) < 0\) before \(x=c\) and \(f'(x) > 0\) after \(x=c\text{,}\) then a local minimum occurs at \(x=c\text{.}\)
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If \(f'(x)\) is the same sign on both sides of \(x=c\text{,}\) then neither a local maximum nor a local minimum occur when \(x=c\text{.}\)
Activity 3.5.8.
Let \(f(x)=x^4-4x^3+4x^2\)
(a)
Find all critical points of \(f(x)\text{.}\) Draw them on the same number line.
(b)
What intervals have been created by subdividing the number line at the critical points?
(c)
Pick an \(x\)-value that lies in each interval. Determine whether \(f'(x)\) is positive or negative at each point.
(d)
(e)
List all local extrema.
Theorem 3.5.9. The Second Derivative Test.
Suppose that \(x=c\) is a critical point of \(f(x)\) and that \(f''(x)\) is continuous at \(x=c\text{.}\)
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If \(f'(c) = 0\) and \(f''(c) < 0 \text{,}\) then a local maximum occurs when \(x=c\text{.}\)
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If \(f'(c) = 0\) and \(f''(c) > 0 \text{,}\) then a local minimum occurs when \(x=c\text{.}\)
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If \(f'(c) = 0\) and \(f''(c) = 0 \text{,}\) then this test is inconclusive.
In case of an inconclusive Second Derivative Test, we need to go back to the First Derivative Test to classify the critical point.
Activity 3.5.10.
Consider the function \(f(x)=-x^{3} + 3 \, x + 4\text{.}\)
(a)
Find the open intervals where \(f(x)\) is increasing or decreasing.
(b)
Find the local extrema of \(f(x)\text{.}\)
Remark 3.5.11. Dealing with discontinuities.
Our previous activity dealt with a function that was continuous for all real numbers. Because of that, we could trust our chart to point out local extrema. Let’s now consider what might happen if a function has any discontinuities.
Activity 3.5.12.
Draw a function that is increasing on the left of \(x=1\text{,}\) discontinuous at \(x=1,\) such that \(f(1)=\displaystyle \lim_{x \to 1^+}f(x)\text{,}\) and decreasing to the right of \(x=1\text{.}\) Does the derivative of \(f(x)\) exist at \(x=1\text{?}\) Does your graph have a local maximum or minimum at \(x=1\text{?}\)
Activity 3.5.13.
Let \(f(x)=\frac{x}{(x-2)^2}\text{.}\)
(a)
Note that \(f(x)\) is not defined for \(x=2\text{.}\) But the function may be increasing on one side of \(x=2\) and decreasing on the other! So we include \(x=2\) on your number line.
(b)
Find all critical points of \(f(x)\text{.}\) Plot them and any discontinuities for \(f(x)\) on the same number line.
(c)
What intervals have been created by subdividing the number line at the critical points and at the discontinuities?
(d)
Pick an \(x\)-value that lies in each interval. Determine whether \(f'(x)\) is positive or negative each point.
(e)
(f)
List all local maxima and local minima.
Activity 3.5.14.
For each of the following functions, find the intervals on which \(f(x)\) is increasing or decreasing. Then identify any local extrema using either the First or Second Derivative Test.
(a)
\(f(x)=x^3+3x^2+3x+1\)
(b)
(c)
\(f(x)=(x^2-9)^{2/3}\)
(d)
\(f(x)= \ln(2x-1)\text{.}\) (Hint: think about the domain of this one before you get started!)
(e)
\(f(x)=\frac{x^2}{x^2-4}\)
Theorem 3.5.15. The Mean Value Theorem.
If \(f\) is continuous and differentiable on the closed interval \([a,b]\text{,}\) then there is some point \(c\) in the interval where \(f'(c)\) is equal to the slope of the secant line through the points \((a,f(a))\) and \((b,f(b))\text{.}\) In symbols, for some \(c\) in \((a,b)\) we have that
\begin{equation*}
f'(c) = \frac{f(b)-f(a)}{b-a}
\end{equation*}
Activity 3.5.16.
(a)
Suppose \(f\) is continuous and differentiable on \([a,b]\) and also suppose that \(f(a)=f(b)\text{.}\) What is the average rate of change of \(f(x)\) on \([a,b]\text{?}\) What does the MVT (Mean Value Theorem) tell you?
(b)
Use part (a) to show with the MVT that \(f(x) = (x-1)^2 + 3\) has a critical point on \([0,2]\text{.}\)
Subsection 3.5.2 Videos
Subsection 3.5.3 Exercises
Exercises available at
https://library.tbil.org/2025/calculus/exercises/#/bank/AD5/
