Labware - MA35 Multivariable Calculus - Single Variable Calculus
 MA35 Labs 1 » Single Variable Calculus Contents1.3 More Derivatives 1.3.1 Second Derivatives 1.3.3 Derivative Test 1.3.4 Taylor Series 1.4 Integration Search

Derivative Test

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The sign of the second derivative can help determine whether a critical point is a maximum or a minimum.

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) = x2, x = 0
• f(x) = x2 - x4, x = 0
• f(x) = x3 - 3x, x = 1
• f(x) = ex - x, x = 0
• f(x) = 1/x2 + 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?