First Derivative Test

Definition:

Let f(x) be a differentiable function and let (a, f(a)) be a critical point of f(x).

1. If f'(a) > 0 to the left of a, and f'(a) < 0 to the right of a, then (a, f(a)) is a local minimum of f(x).

2. If f'(a) < 0 to the left of a, and f'(a) > 0 to the right of a, then (a, f(a)) is a local maximum of f(x).

3. In any other case, the critical point is neither a maximum or minimum. We call it a saddle point.

Example: Let f(x) = 3x^5 - 5x^2. Find all critical points of f(x) and use the first derivative test to classify each point.

Solution:  First, we find the derivative: f'(x) = 15x^4 - 15x^2. We set the derivative equal to zero to find the critical points:

FirstD2So, the critical points are (−1,2), (0,0), and (1,−2). We apply the first derivative test by finding the derivatives of points to the left and the right of each critical point:

FirstD3So what does this tell us? Well we look at each point individually.

For (-1, 2), the derivative changes from positive to negative. This tells us that (-1, 2) is a local maximum.

For (0,0), the derivative is negative both to the left and to the right of the point. This tells us that (0,0) is a saddle point.

For (1, -2), the derivative changes from negative to positive. This tells us that (1, -2) is a local minimum.

We can examine the graph and confirm our findings:

FirstD4

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