Learning Outcomes
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Determine the intervals of concavity of a twice differentiable function and find all of its points of inflection.
Section 3.6 Concavity and Inflection (AD6)
Subsection 3.6.1 Activities
Observation 3.6.1.
In addition to asking whether a function is increasing or decreasing, it is also natural to inquire how a function is increasing or decreasing. Activity 3.6.2 describes three basic behaviors that an increasing function can demonstrate on an interval, as pictured in Figure 3.6.3
Activity 3.6.2.
Sketch a sequence of tangent lines at various points to each of the following curves in Figure 3.6.3.
(a)
Look at the curve pictured on the left of Figure 3.6.3. How would you describe the slopes of the tangent lines as you move from left to right?
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The slopes of the tangent lines decrease as you move from left to right.
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The slopes of the tangent lines remain constant as you move from left to right.
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The slopes of the tangent lines increase as you move from left to right.
(b)
Look at the curve pictured in the middle of Figure 3.6.3. How would you describe the slopes of the tangent lines as you move from left to right?
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The slopes of the tangent lines decrease as you move from left to right.
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The slopes of the tangent lines remain constant as you move from left to right.
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The slopes of the tangent lines increase as you move from left to right.
(c)
Look at the curve pictured on the right of Figure 3.6.3. How would you describe the slopes of the tangent lines as you move from left to right?
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The slopes of the tangent lines decrease as you move from left to right.
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The slopes of the tangent lines remain constant as you move from left to right.
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The slopes of the tangent lines increase as you move from left to right.
Remark 3.6.4.
On the leftmost curve in Figure 3.6.3, as we move from left to right, the slopes of the tangent lines will increase. Therefore, the rate of change of the pictured function is increasing, and this explains why we say this function is increasing at an increasing rate.
Observation 3.6.5.
We must be extra careful with our language when dealing with negative numbers. For example, it can be tempting to say that “\(-100\) is bigger than \(-2\text{.}\)” But we must remember that “greater than” describes how numbers lie on a number line: \(-100\) is less than \(-2\) becomes it comes earlier on the number line. It might be helpful to say that “\(-100\) is “more negative” than \(-2\text{.}\)”
Activity 3.6.6.
Sketch a sequence of tangent lines at various points to each of the following curves in Figure 3.6.7.
Diagram Exploration Keyboard Controls
(a)
Look at the curve pictured on the left in Figure 3.6.7. How would you describe the slopes of the tangent lines as you move from left to right?
-
The slopes of the tangent lines decrease as you move from left to right.
-
The slopes of the tangent lines remain constant as you move from left to right.
-
The slopes of the tangent lines increase as you move from left to right.
(b)
Look at the curve pictured in the middle in Figure 3.6.7. How would you describe the slopes of the tangent lines as you move from left to right?
-
The slopes of the tangent lines decrease as you move from left to right.
-
The slopes of the tangent lines remain constant as you move from left to right.
-
The slopes of the tangent lines increase as you move from left to right.
(c)
Look at the curve pictured on the right in Figure 3.6.7. How would you describe the slopes of the tangent lines as you move from left to right?
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The slopes of the tangent lines decrease as you move from left to right.
-
The slopes of the tangent lines remain constant as you move from left to right.
-
The slopes of the tangent lines increase as you move from left to right.
Remark 3.6.8.
Recall the terminology of concavity: when a curve bends upward, we say its shape is concave up. When a curve bends downwards, we say its shape is concave down.
Activity 3.6.9.
Look at the curves in Figure 3.6.10. Which curve is concave up? Which one is concave down? Why? Try to explain using the graph!
Definition 3.6.11.
Let \(f\) be a differentiable function on some interval \((a,b)\text{.}\) Then \(f\) is concave up on \((a,b)\) if and only if \(f'\) is increasing on \((a,b)\text{;}\) \(f\) is concave down on \((a,b)\) if and only if \(f'\) is decreasing on \((a,b)\text{.}\)
Activity 3.6.12.
Look at how the slopes of the tangent lines change from left to right for each of the two graphs in Figure 3.6.10
(a)
Look at the curve pictured on the left in Figure 3.6.10. How would you describe the slopes of the tangent lines as you move from left to right?
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The slopes of the tangent lines decrease as you move from left to right.
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The slopes of the tangent lines increase as you move from left to right.
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The slopes of the tangent lines go from increasing to decreasing as you move from right to left.
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The slopes of the tangent lines go from decreasing to increasing as you move from right to left.
(b)
Which of the following statements is true about the function on the left in Figure 3.6.10?
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\(f'(x) > 0 \) on the entire interval shown.
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\(f'(x) < 0 \) on the entire interval shown.
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\(f''(x) > 0 \) on the entire interval shown.
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\(f''(x) < 0 \) on the entire interval shown.
(c)
Look at the curve pictured on the right in Figure 3.6.10. How would you describe the slopes of the tangent lines as you move from left to right?
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The slopes of the tangent lines decrease as you move from left to right.
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The slopes of the tangent lines increase as you move from left to right.
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The slopes of the tangent lines go from increasing to decreasing as you move from right to left.
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The slopes of the tangent lines go from decreasing to increasing as you move from right to left.
(d)
Which of the following statements is true about the function on the right in Figure 3.6.10?
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\(f'(x) > 0 \) on the entire interval shown.
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\(f'(x) < 0 \) on the entire interval shown.
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\(f''(x) > 0 \) on the entire interval shown.
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\(f''(x) < 0 \) on the entire interval shown.
Theorem 3.6.13. Test for Concavity.
Suppose that \(f(x)\) is twice differentiable on some interval \((a,b)\text{.}\) If \(f'' > 0\) on \((a,b)\text{,}\) then \(f\) is concave up on \((a,b)\text{.}\) If \(f'' < 0\) on \((a,b)\text{,}\) then \(f\) is concave down on \((a,b)\text{.}\)
Observation 3.6.14.
In the previous section, we saw in Activity 3.5.8 how to use critical points of the function and the sign of the first derivative to identify intervals of increase/decrease of a function. The next activity Activity 3.6.15 uses the critical points of a first derivative function and the sign of its second derivative (accordingly to Theorem 3.6.13) to identify where the original function is concave up/down.
Activity 3.6.15.
Let \(f(x)=x^4-54x^2\text{.}\)
(a)
Find all the zeros of \(f''(x)\text{.}\)
(b)
What intervals have been created by subdividing the number line at zeros of \(f''(x)\text{?}\)
(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 the intervals where \(f(x)\) is concave up and all the intervals where \(f(x)\) is concave down.
Definition 3.6.16.
If \(x=c\) is a point where \(f''(x)\) changes sign, then the concavity of graph of \(f(x)\) changes at this point and we call \(x=c\) an inflection point of \(f(x)\text{.}\)
Activity 3.6.17.
Use the results from Activity 3.6.15 to identify all of the inflection points of \(f(x)=x^4-54x^2\text{.}\)
Activity 3.6.18.
For each of the following functions, describe the open intervals where it is concave up or concave down, and any inflection points.
(a)
\(f(x)=-\frac{1}{4} \, x^{5} - \frac{5}{2} \, x^{4} - \frac{15}{2} \, x^{3}\)
(b)
\(f(x)=\frac{3}{20} \, x^{5} + x^{4} - \frac{5}{2} \, x^{3}\)
(c)
Activity 3.6.19.
Consider the following table. The values of the first and second derivatives of \(f(x)\) are given on the domain \([0,7]\text{.}\) The function \(f(x)\) does not suddenly change behavior between the points given, so the table gives you enough information to completely determine where \(f(x)\) is increasing, decreasing, concave up, and concave down.
\begin{equation*}
\begin{array}{c|cccccccc}
x
& 0
& 1
& 2
& 3
& 4
& 5
& 6
& 7
\\\hline
f'(x)
& 2
& 0
& -2
& 0
& 2
& 1
& 0
& -1
\\\hline
f''(x)
& -2
& -1
& 0
& 1
& 0
& -1
& 0
& 3
\\
\end{array}
\end{equation*}
(a)
List all the critical points of \(f(x)\) that you can find using the table above.
(b)
Use the First Derivative Test to classify the critical numbers (decide if they are a max or min). Write full sentence stating the conclusion of the test for each critical number.
(c)
On which interval(s) is \(f(x)\) increasing? On which interval(s) is \(f(x)\) decreasing? List all the critical points of \(f(x)\) that you can find using the table above.
(d)
There is one critical number for which the Second Derivative Test is inconclusive. Which one? You can still determine if it is a max or min using the First Derivative Test!
(e)
List all the critical points of \(f'(x)\) that you can find using the table above.
(f)
(g)
List all the inflection points of \(f(x)\) that you can find using the table above.
Subsection 3.6.2 Videos
See Figure 3.5.17.
Subsection 3.6.3 Exercises
Exercises available at
https://library.tbil.org/2026/calculus/exercises/#/bank/AD6/
