Derivative Test
Text The sign of the second derivative can help determine whether a critical point is a maximum or a minimum.
If f'(x_{0}) = 0 and f''(x_{0}) > 0, then the function graph is convex upward at that point and the point must be a relative minimum. Similarly if f'(x_{0}) = 0 and f''(x_{0}) < 0, then the function graph is convex downward at that point and the point must be a relative maximum. If f'(x_{0} = 0 and f''(x_{0}) = 0, then this test is inconclusive. If f''(x_{0}) > 0 on one side of x_{0} in a neighborhood of x_{0} and f''(x_{0}) < 0 on the other side, then x_{0} is called a horizontal inflection point.
Demos
Derivative Test
 
This demo displays three curves. The first, in gray, is a graph of the function f(x). The second graphs f '(x) and is red when f(x) is increasing and green when f(x) is decreasing. The third graphs f ''(x) and is magenta when f(x) is concave downward and yellow when f ''(x) when f(x) is concave upward. Notice what happens to each of these functions at maxima, minima, and inflection points.

Exercises 1. Use the second derivative test to determine whether each of the following critical points is a maximum or a minimum, and check using the demo:
 f(x) = x^{2}, x = 0
 f(x) = x^{2}  x^{4}, x = 0
 f(x) = x^{3}  3x, x = 1
 f(x) = e^{x}  x, x = 0
 f(x) = 1/x^{2} + x, x = ^{4}√2
2. Describe what the graphs of the function, derivative, and second derivative look like at local maxima. What about local minima?
