Extreme Value Theorem

Definition:  One of the most useful applications of the derivative is finding maxima and minima of functions. A local maximum is the point with the highest function value of other points around it; and a local minimum is a point with the lowest function value of points around it. A global maximum is the point with the highest function value over the entire domain, and the minimum is the point with the lowest function value over the entire domain.

Take a look at the below image. Look at the function at the minimum and the maxima. The derivative in both places is 0! In fact, this will be true for all maxima and minima for a function: if a function has a minimum or a maximum at a point, the derivative there will be 0. Unfortunately, the converse is not true: if the derivative is 0, this does not necessarily signal a maximum or a minimum. It could signal a saddle point, as in the image above. In any case, a point where a function’s derivative is zero is called a critical point of the function.

minmaxsaddle

So, how do we know if we have a maximum or a minimum, or neither? One method involves looking at how the function is changing around the critical point. If our point is a maximum, then the function should be increasing right before the point, and decreasing right after the point. If our point is a minimum, then the function should be decreasing right before the point, and increasing right after.

Recall that we can look at the derivative to determine whether the function is increasing or decreasing: if the derivative is positive, then the function is increasing; and if the derivative is negative, then the function is decreasing. Keeping all of this in mind, we can now formulate the first derivative test:

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 $latex f(x) 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:

FirstD2

So, 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:

FirstD3

So 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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