Concavity & Inflection Points Worksheet 9

Use derivative signs and stated geometric constraints to interpret functions and rates.

hard: Use another set to practice explaining and checking your answers.

23 problems · 25-30 minutes · Skills: derivatives, concavity, calculus

About This Worksheet

This hard-level worksheet contains 23 problems designed for 12th Grade students practicing concavity & inflection points. Hard worksheets feature challenging problems including multi-step questions, missing values, and real-world applications to push students further.

Manipulate algebraic expressions and understand functions, limits and derivative notation.

Use derivative signs and stated geometric constraints to interpret functions and rates.
Check your result using the method in the example before moving to the next question.

How to Solve: Worked Example

Example Problem: For f(x) = (x - 3)³ - 2, find the concavity intervals and inflection point.

Solution Steps:

f″(x) = 6(x - 3).
Concavity changes where the second derivative changes sign.

Answer: Concave down for x < 3; concave up for x > 3. Inflection point (3, -2).

Tip: Check your result using the method in the example before moving to the next question.

Paper Include separate answer key in PDF

For f(x) = (x + 4)³ + 5, find the concavity intervals and inflection point.

For f(x) = (x + 1)³ - 2, find the concavity intervals and inflection point.

For f(x) = (x - 4)³ - 5, find the concavity intervals and inflection point.

For f(x) = (x - 4)³ + 4, find the concavity intervals and inflection point.

For f(x) = (x + 1)³ - 5, find the concavity intervals and inflection point.

For f(x) = (x + 5)³ - 1, find the concavity intervals and inflection point.

For f(x) = (x - 1)³ + 3, find the concavity intervals and inflection point.

For f(x) = (x + 1)³ + 4, find the concavity intervals and inflection point.

For f(x) = (x + 3)³ - 1, find the concavity intervals and inflection point.

For f(x) = (x - 3)³ + 2, find the concavity intervals and inflection point.

For f(x) = (x - 1)³ + 5, find the concavity intervals and inflection point.

For f(x) = (x - 1)³ - 5, find the concavity intervals and inflection point.

For f(x) = (x - 3)³ - 4, find the concavity intervals and inflection point.

For f(x) = (x - 4)³ + 0, find the concavity intervals and inflection point.

For f(x) = (x - 2)³ - 4, find the concavity intervals and inflection point.

For f(x) = (x - 2)³ + 4, find the concavity intervals and inflection point.

For f(x) = (x + 2)³ + 0, find the concavity intervals and inflection point.

For f(x) = (x + 2)³ - 4, find the concavity intervals and inflection point.

For f(x) = (x - 3)³ - 2, find the concavity intervals and inflection point.

For f(x) = (x - 4)³ - 4, find the concavity intervals and inflection point.

For f(x) = (x + 1)³ - 3, find the concavity intervals and inflection point.

For f(x) = (x - 4)³ - 3, find the concavity intervals and inflection point.

For f(x) = (x + 2)³ + 4, find the concavity intervals and inflection point.

Answer Key

1. Concave down for x < -4; concave up for x > -4. Inflection point (-4, 5).2. Concave down for x < -1; concave up for x > -1. Inflection point (-1, -2).3. Concave down for x < 4; concave up for x > 4. Inflection point (4, -5).4. Concave down for x < 4; concave up for x > 4. Inflection point (4, 4).5. Concave down for x < -1; concave up for x > -1. Inflection point (-1, -5).6. Concave down for x < -5; concave up for x > -5. Inflection point (-5, -1).7. Concave down for x < 1; concave up for x > 1. Inflection point (1, 3).8. Concave down for x < -1; concave up for x > -1. Inflection point (-1, 4).9. Concave down for x < -3; concave up for x > -3. Inflection point (-3, -1).10. Concave down for x < 3; concave up for x > 3. Inflection point (3, 2).11. Concave down for x < 1; concave up for x > 1. Inflection point (1, 5).12. Concave down for x < 1; concave up for x > 1. Inflection point (1, -5).13. Concave down for x < 3; concave up for x > 3. Inflection point (3, -4).14. Concave down for x < 4; concave up for x > 4. Inflection point (4, 0).15. Concave down for x < 2; concave up for x > 2. Inflection point (2, -4).16. Concave down for x < 2; concave up for x > 2. Inflection point (2, 4).17. Concave down for x < -2; concave up for x > -2. Inflection point (-2, 0).18. Concave down for x < -2; concave up for x > -2. Inflection point (-2, -4).19. Concave down for x < 3; concave up for x > 3. Inflection point (3, -2).20. Concave down for x < 4; concave up for x > 4. Inflection point (4, -4).21. Concave down for x < -1; concave up for x > -1. Inflection point (-1, -3).22. Concave down for x < 4; concave up for x > 4. Inflection point (4, -3).23. Concave down for x < -2; concave up for x > -2. Inflection point (-2, 4).

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