[list][*]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[/*][/list]

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. [br][br]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. [br][br]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.