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.

[b]Concavity and Inflection Points: [/b]The second derivative [math]f''[/math] gives us information about both the first derivative [math]f'[/math] and the original function [math]f[/math]. [br][list][*][math]f[/math] is [b]concave up[/b] [math]\Leftrightarrow[/math] [math]f'[/math] is increasing [math]\Leftrightarrow[/math] [math]f''[/math] is positive[/*][*][math]f[/math] is [b]concave down[/b] [math]\Leftrightarrow[/math] [math]f'[/math] is decreasing [math]\Leftrightarrow[/math] [math]f''[/math] is negative[/*][/list]An[b] inflection point[/b] is a point where a function changes from concave up to concave down, or vice versa. Potential inflection points are points where [math]f''(c)=0[/math] or [math]f''(c)[/math] does not exist. [br][br][b]The Second Derivative Test: [/b]The second derivative test is an alternative way to test the critical points of f to determine whether f has a local extreme value there. If [math]x=c[/math] is a critical point where [math]f'(c)=0[/math], then:[br][list][*]If [math]f''(c)<0[/math], then f is concave down (i.e., f' is decreasing) and f has a local maximum at x = c. [/*][*]If [math]f''(c)>0[/math], then f is concave up (i.e., f' is increasing) and f has a local minimum at x = c. [/*][*]If [math]f''(c)=0[/math], 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. [/*][/list]