5-C Second Derivative Test
Instructions
Use the input boxes to define a function f(x) on an interval [a, b]. Use the checkboxes for a and b to include/exclude the endpoints of the interval. Use the checkboxes to show/hide interior extreme values, critical points, intervals of increase/decrease (monotonicity), and the graphs of the first and second derivative functions.
5-C Second Derivatives and Concavity
Concavity and Inflection Points: The second derivative gives us information about both the first derivative and the original function .
- is concave up is increasing is positive
- is concave down is decreasing is negative
- If , then f is concave down (i.e., f' is decreasing) and f has a local maximum at x = c.
- If , then f is concave up (i.e., f' is increasing) and f has a local minimum at x = c.
- If , then f is neither concave up nor concave down (i.e., f' is neither increasing nor decreasing) and the second derivative test is inconclusive. Use the first derivative test instead.