6C. Second Derivatives & Function Behavior
Instructions:
- Use the input box to define the function y=f(x). Use the input boxes for a and b to define the domain interval and use the checkboxes for a and b to include/exclude the endpoints for the domain.
- Use the Critical Points and Interior Extrema checkboxes to show/hide the critical points and interior extreme values, respectively.
- Use the checkboxes for f' and Monotonicity to show/hide the graph of the first derivative and to highlight where f is increasing or decreasing, respectively.
- Use the Inflection Points checkbox to show/hide inflection points.
- Use the checkboxes for f'' and Concavity to show/hide the graph of the second derivative and to highlight where f is concave up or concave down, respectively
Second Derivatives and Function Behavior
The first derivative f' gives information about the monotonicity of f. The critical points of f occur where the first derivative is zero (i.e., f'(x) = 0) and represent the locations where f can change its monotonicity (e.g., from increasing to decreasing). Because of this the first derivative is ideal for finding local maximum and minimum values.
The second derivative f'' gives information about where f' is increasing or decreasing, which translates to information about where f is concave up or down (by definition of concavity). The inflection points of f are the points where f changes concavity, which can only happen when f''(x) = 0.
Note that a change in concavity of f is the same as a change in monotonicity of f'. Therefore, the inflection points of f are local maximum/minimum values of f'. In other words, inflection points represent points of (local) maximum/minimum steepness on the graph of f.