Critical points are places where a function may have a local maximum, a local minimum, or neither. In calculus, the first derivative test and the second derivative test are two main tools for classifying these points. They matter because they connect the shape of a graph to rates of change and concavity.
This helps in optimization, curve sketching, and interpreting real situations such as cost, motion, and growth.
Key Facts
- Critical points occur where f'(c) = 0 or f'(c) is undefined, as long as f(c) exists.
- First Derivative Test: if f' changes from positive to negative at c, then f has a local maximum at c.
- First Derivative Test: if f' changes from negative to positive at c, then f has a local minimum at c.
- First Derivative Test: if f' does not change sign at c, then f has neither a local maximum nor a local minimum at c.
- Second Derivative Test: if f'(c) = 0 and f''(c) < 0, then f has a local maximum at c.
- Second Derivative Test: if f'(c) = 0 and f''(c) > 0, then f has a local minimum at c, but if f''(c) = 0, the test is inconclusive.
Vocabulary
- Critical point
- A critical point is an x-value in the domain of f where f'(x) = 0 or f'(x) is undefined.
- Local maximum
- A local maximum is a point where the function value is greater than or equal to nearby function values.
- Local minimum
- A local minimum is a point where the function value is less than or equal to nearby function values.
- First derivative test
- The first derivative test classifies a critical point by checking whether f'(x) changes sign around it.
- Second derivative test
- The second derivative test classifies a critical point by using the sign of f''(x) to determine concavity at that point.
Common Mistakes to Avoid
- Using the second derivative test when f'(c) is not zero. The usual second derivative test for extrema requires f'(c) = 0 before using the sign of f''(c).
- Assuming f''(c) = 0 means there is no extremum. This is wrong because f''(c) = 0 only makes the second derivative test inconclusive, so another test is needed.
- Forgetting to check both sides of a critical point in the first derivative test. A single value of f'(x) does not show whether the derivative changes sign.
- Calling every flat point a maximum or minimum. A point with f'(c) = 0 can be a flat inflection point, such as on f(x) = x^3 at x = 0.
Practice Questions
- 1 For f(x) = x^2 - 6x + 5, find the critical point and classify it using the second derivative test.
- 2 For f(x) = x^3 - 3x, find all critical points and classify each one using the first derivative test.
- 3 A function has f'(2) = 0 and f''(2) = 0. Explain why the second derivative test is inconclusive and describe what information the first derivative test would need.