Use derivative signs and stated geometric constraints to interpret functions and rates.
Before you begin
Manipulate algebraic expressions and understand functions, limits and derivative notation.
Worked example
For f(x) = x³ − 27x, find the local maximum, local minimum, inflection point and end behavior to guide a sketch.
f′(x) = 3x² − 27, with critical points ±3.
Check the derivative signs for max/min.
f″(x) = 6x changes sign at zero. The leading term is a positive cubic.
Answer: Local maximum (−3, 54); local minimum (3, −54); inflection (0, 0). Left end down; right end up.
Check your result using the method in the example before moving to the next question.
Choose your level
These sets use the same exercise types and number ranges across level labels. Choose a fresh set for additional practice.
Easy: Start with the preview and follow the worked example.
Medium: Practice the same exercise types with a fresh numbered set.
Hard: Use another set to practice explaining and checking your answers.
Questions and answer calculations checked on 2026-09-17. How we check worksheets. Free for personal and classroom use.
Preview and print an easy set
Try six questions below, or choose a full set at your level.
1
For f(x) = x³ − 12x, find the local maximum, local minimum, inflection point and end behavior to guide a sketch.
2
For f(x) = x³ − 75x, find the local maximum, local minimum, inflection point and end behavior to guide a sketch.
3
For f(x) = x³ − 48x, find the local maximum, local minimum, inflection point and end behavior to guide a sketch.
4
For f(x) = x³ − 108x, find the local maximum, local minimum, inflection point and end behavior to guide a sketch.
5
For f(x) = x³ − 3x, find the local maximum, local minimum, inflection point and end behavior to guide a sketch.
6
For f(x) = x³ − 27x, find the local maximum, local minimum, inflection point and end behavior to guide a sketch.
Answer Key
1. Local maximum (−2, 16); local minimum (2, −16); inflection (0, 0). Left end down; right end up.2. Local maximum (−5, 250); local minimum (5, −250); inflection (0, 0). Left end down; right end up.3. Local maximum (−4, 128); local minimum (4, −128); inflection (0, 0). Left end down; right end up.4. Local maximum (−6, 432); local minimum (6, −432); inflection (0, 0). Left end down; right end up.5. Local maximum (−1, 2); local minimum (1, −2); inflection (0, 0). Left end down; right end up.6. Local maximum (−3, 54); local minimum (3, −54); inflection (0, 0). Left end down; right end up.