# First derivative test

The first derivative test is used to examine where a function is increasing or decreasing on its domain and to identify its local maxima and minima.

The first derivative is the slope of the line tangent to the graph at a given point. It may be helpful to think of the first derivative as the slope of the graph. When the slope is positive, the graph is increasing. When it is negative, the graph is decreasing. When the slope is 0, the point is a critical point, and may be a local maximum or minimum.

Given a differentiable function, the first derivative test can be used to find any local minima or maxima of the function through the following steps:

1. Differentiate the function.
2. Set the derivative of the function equal to 0 and solve the equation to find any critical points.
3. Test values before and after the critical points to determine whether the function is increasing (positive derivative) or decreasing (negative derivative) around the point.

Then, note that:

• If the first derivative changes from positive to negative at the given point, then the point is a local maximum.
• If the first derivative changes from negative to positive at the given point, then the point is a local minimum.
• If the first derivative does not change at the given point, then the point is not a local minimum or maximum.

Example:

Given the function , find the critical points and any local minima or maxima of the function.

Using the steps above,

Step 1: Differentiate f(x) Step 2: Solve for x   Thus, the first derivative, x3 - 8, has a critical point at x = 2.

Step 3: Test points around the critical point, such as x = 1 and x = 3:

For x = 1, f '(x) = 13 - 8 = 1 - 8 = -7

For x = 3, f '(x) = 33 - 8 = 27 - 8 = 19

Since f'(1) and f'(3), the test points around our critical point, change from negative to positive, this indicates a negative slope on the graph of f(x) before the critical point, and a positive slope after the critical point (from left to right). The critical point x = 2 is thus a local minimum, as can be seen in the graph of f(x) provided below.