Second Derivatives & Concavity
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.
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.