Critical numbers where the derivative of the function equals zero locate relative minima, relative maxima, and points of inflection of a function. To find these critical numbers, you take the derivative of the function, set it equal to zero, and solve for x (or whatever the independent variable happens to be). It is easy to construct a function that will give you these critical numbers automatically. The function is:
Define critnum(f, x) = solve(d(f,x)=0,x)
(If you don't remember how to enter a user-defined function, click here.)
The middle picture shows a (partial) solution for Example 2 on p. 158 of the Larsen text. The critical numbers for the function f(x) = 3x4 - 4x3 are x = 0 and x = 1. The graph on the right (the window is -1 < x < 2, -2 < y < 10) shows that x = 0 produces a point of inflection, while x = 1 produces a minimum value. |
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