The First Derivative Test

Using the First Derivative to Classify Critical Points

The First Derivative Test

Suppose \(c\) is a critical point of the continuous function \(f\).

1. If \(f'(x)\) changes sign from positive to negative at \(x=c\), then \(f(c)\) is a relative maximum.

2. If \(f'(x)\) changes sign from negative to positive at \(x=c\), then \(f(c)\) is a relative minimum.

3. If \(f'(x)\) does not change sign at \(x=c\), then \(f(c)\) is not a relative extrema.

Example 1

In Example 1, we found that the critical points of

\[f(x) = \sqrt[3]{x^2-1}\]

were \(x=-1\), \(x=0\), and \(x=1\). Classify each critical point using the First Derivative Test.

Step 1:   Break up the domain of   \(f'(x)\)   at each critical point.

Plug in one number from each subinterval into \(f'(x)\) to determine the sign of \(f'(x)\) on each interval.

We have \(f'(x) = \dfrac{2x}{3(x^2 - 1)^{2/3}}\).

Fig. 4 Interval analysis of \(f'(x) = \dfrac{2x}{3(x^2 - 1)^{2/3}}\)

Long Text Description

A number line with positive and negative signs assigned to intervals, which are negative to the left of negative one, negative between one and zero, positive between zero and one, and positive to the right of one.

Step 2:   Classify each critical point.

Since \(f'(x)\) changes from negative to positive at \(x=0\), \(f(x)\) has a relative minimum at \(x=0\).

Since \(f'(x)\) does not change sign at \(x=-1\) and \(x=1\), \(f(x)\) does not have relative extrema at these values of \(x\).

Example 2

In Example 2, we found that the critical points of

\[f(x) = x^3 +3x^2 - 24x + 1\]

were \(x=-4\) and \(x=2\). Classify the critical point using the First Derivative Test.

Step 1:   Break up the domain of   \(f'(x)\)   at each critical point.

Plug in one number from each subinterval into \(f'(x)\) to determine the sign of \(f'(x)\) on each interval.

We have \(f'(x) = 3(x+4)(x-2)\).

Fig. 5 Interval analysis of \(f'(x) = 3(x+4)(x-2)\).

Long Text Description

A number line with positive and negative signs assigned to intervals, with positive to the left of negative four, negative from negative four and two, and positive to the right of two.

Step 2:   Classify each critical point.

Since \(f'(x)\) changes from positive to negative at \(x=-4\), \(f(x)\) has a relative maximum at \(x=-4\). Since \(f'(x)\) changes from negative to positive at \(x=2\), \(f(x)\) has a relative minimum at \(x=2\).