Derivative functions contain important information. Let’s look at a couple of examples. Going back to our running example, *f(t)* describes a runner’s position as a function of time; therefore, the derivative function *f’(t)* describes the runner’s speed at any time *t*. Let’s say we have a function *s(t)* that describes the real-time sales of a particular product; the corresponding derivative function *s’(t)* describes how those sales are changing in real-time.

Let’s look at the general information we can get as well. If *f’(x)* > 0, the corresponding function *f(x)* is increasing. Keep in mind that *f(x)* itself may be negative even though the derivative *f’(x)* is positive. The derivative says nothing about the value of the original function, only about how it is changing. Similarly, if *f’(x)* < 0, then the original function is decreasing.

If *f’(x)* = 0, then the function is not changing. This can mean one of a few things.

- It may mean that the function has reached a
*local maximum*(or maxima). A local maximum is a value of x where*f’(x)*changes from positive to negative and thus hits 0 along the way. In*f(x)*, the local maximum is higher than all the points around it. - It may also mean that the function has reached what is called a
*local minimum*. A local minimum is lower than the points around it. When*f’(x)*goes from negative values to 0 to positive values, a local minimum forms. - It may be an
*inflection point*. This is a point where a function has a change in the direction of curvature. For example, the curve of the function goes from “facing down” to “facing up.” Finding inflection points involves a second derivative test, which we will not get to in this lesson.

*Global maxima* and *global minima* are the largest or smallest over the entire range of a function.

In machine learning models, maxima and minima are key concepts as they’re usually what we are trying to find when we optimize!

### Instructions

The animation to the right outlines our newfound knowledge of extrema. The first image shows a part of a function that is increasing. What do we know about the value of *f’(x)* in this section of *f(x)*? Click below to check your answer.

## Solution

*f'(x) > 0*because the slope of the function is positive.

In the second image, we see a global maximum. This is the largest value over the entire range of the function. What do we know about the value of *f’(x)* at this point?

## Solution

*f'(x) = 0*because the slope of the function is going from positive to negative and at the global maximum it is equal to zero,

In the third image, we see a part of the function that is decreasing. What do we know about the value of *f’(x)* at this point?

## Solution

*f'(x) < 0*because the slope of the function is negative.

In the fourth image, we see a local minimum. This is the smallest value over a specific range of the function. What do we know about the value of *f’(x)* at this point?

## Solution

*f'(x) = 0*because the slope of the function is going from negative to positive and at the local minimum it is equal to zero,

In the final image, we see a local maximum. This is the largest value over a specific range of the function. What do we know about the value of *f’(x)* at this point?

## Solution

*f'(x) = 0*because the slope of the function is going from positive to negative and at the local maximum it is equal to zero,