Critical values of a cubic: (Jump directly to the Test Problem)
Sample problem:
The extrema or critical values are located where the derivative of f(x) is zero. We can use the Power, Scalar Multiple, and Sum Rules to get
Setting this to 0 and factoring, we get
Since f ' is defined for all values of x, the critical numbers are -3 and +2.
Next we wish to use the First Derivative Test to determine the slope of the graph of f(x) in the 3 intervals defined by these two values as shown in the following table:
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Interval |
To left of -3 |
Between -3 and 2 |
To the right of 2 |
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Point in interval |
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Sign of f '(x) at point |
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Character of graph of f(x) at point |
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Sketch |
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First choose a point in each interval, near a boundary of the interval: