Calculus 5.

6 Determining Concavity Notes

Write your questions
and thoughts here!
What is concavity? The state or quality of being concave.
Concave up: Concave down:

If a function is concave up (like a parabola), what is 𝑓 is doing.

If 𝑓 is concave up, then 𝑓 is increasing. If 𝑓 is concave down, then 𝑓 is decreasing.

This leads us to the following…

𝑓 0 means 𝑓 is concave up. 𝑓 0 means 𝑓 is concave down.

y
𝒇 𝒙 𝐬𝐢𝐧 𝒙


x
   
   
   


1. Find the intervals of concavity for 𝑓 𝑥 𝑥 6𝑥 𝑥 3.

Point of Inflection
There is a point of inflection of 𝑓 at 𝑥 𝑐 if 𝑓 𝑐 is defined and 𝑓 changes signs at 𝑥 𝑐.

In other words, a point of inflection is where the graph changes concavity.

Write your questions
and thoughts here!
Two common mistakes when finding a point of inflection
1. Assuming that 𝒇 𝟎 means there is a point of inflection.
2. Assuming that 𝒇 𝟎 means there is no point of inflection.

2. Given the graph of 𝒇 , find the points of 3. Given the graph of 𝒇 , find the points of
inflection and state the intervals of inflection and state the intervals of
concavity. concavity.

𝒇 𝒙
𝒇 𝒙

4. Does the line tangent to the graph of 𝑓 𝑥 𝑥𝑒 at 𝑥 1 lie above or below the graph
of 𝑓? Why?

5.6 Determining Concavity

Calculus
Practice

1. 𝟏 𝟏 𝟏
x 𝟑 𝒙
𝟐 𝟐 𝟐
𝒙 𝟑

𝒈′′ 𝒙 Positive 𝟎 Negative

Use the table above to find the following.

Intervals where 𝑔 𝑥 is concave up: Intervals where 𝑔 𝑥 is concave Point(s) of Inflection:
down:
Find the point(s) of inflection for each function. Justify your answer.
2. 𝑓 𝑥 sin on the interval 𝜋, 3𝜋 3. 𝑓 𝑥 𝑒

4. ℎ 𝑥 2𝑥 5 5. 𝑓 𝑥 2𝑥 8𝑥 3

State the intervals of concavity and justify your answer.

6. 𝑔 𝑥 7. 𝑓 𝑥 𝑥 12𝑥
The graph of 𝒇 𝒙 is shown. Find the point(s) of inflection.
8. 9. 10.
𝒇 𝒙
𝒇 𝒙
𝒇 𝒙

The graph of 𝒇 𝒙 is shown. Find the point(s) of inflection.

11. 12. 13.
4 4
𝒇 𝒙 3
𝒇 𝒙 3
𝒇 𝒙
2 2

1 1

4 3 2 1 1 2 3 4 4 3 2 1 1 2 3 4
1 1

2 2

3 3

Does the line tangent to the graph of 𝒉 at the given value of 𝒙 lie above or below the graph of 𝒉? Why?
14. ℎ 𝑥 2𝑥 4𝑥 3𝑥 at 𝑥 2 15. ℎ′ 𝑥 at 𝑥 2
5.6 Determining Concavity Test Prep
16. Calculator active problem. Let 𝑓 𝑥 sin 𝑥 . Which of the following three statements are true?

I. 𝑓 is concave up on 0, 1.77 and 2.51, 3.06 .

II. 𝑓 is concave down on 1.78, 2.50 .
III. 𝑓 is increasing on 0, 1.77 .

(A) I and II only (D) II and III only

(B) I and III only (E) III only

(C) I, II, and III

17. Consider the differential equation 4𝑥 𝑦. Find . Determine the concavity of all solution
curves for the given differential equation in Quadrant I. Give a reason for your answer.

18. Write an equation of the line tangent to 𝑦 𝑥 3𝑥 4 at its point of inflection.

19. If the graph of 𝑦 𝑥 𝑎𝑥 𝑏𝑥 4 has a point of inflection at 1, 6 , what is the value of 𝑏?

(A) 3

(B) 0

(C) 1

(D) 3

(E) It cannot be determined from the information given.