Second derivative test

Definition

The second derivative test helps determine the local extrema (maximum or minimum points) of a function. It uses the sign of the second derivative at critical points to assess concavity.

5 Must Know Facts For Your Next Test

1. The second derivative test requires finding the critical points where the first derivative is zero or undefined.
2. If $f''(c) > 0$, then $f$ has a local minimum at $x = c$.
3. If $f''(c) < 0$, then $f$ has a local maximum at $x = c$.
4. If $f''(c) = 0$, the test is inconclusive, and other methods must be used to determine whether there is a local extremum.
5. The second derivative test can help identify points of inflection when combined with an analysis of concavity.

Review Questions

• What does it mean if the second derivative at a critical point is positive?
• What steps are required to apply the second derivative test to find local extrema?
• How would you proceed if the second derivative at a critical point is zero?

Related terms

First Derivative: The first derivative of a function represents its rate of change or slope.

Concavity: Concavity describes whether a graph curves upwards or downwards, determined by the sign of the second derivative.

Critical Point: A point on a graph where the first derivative is zero or undefined, potentially indicating a local extremum.