Here is the graph of a function f.

Consider an interval along the function curve. The greatest value of the function in that interval, f(c), is called the relative maximum or the local maximum.

When x is near c, a function f has a relative maximum at x = c if f(c) is greater than f(x) for all values of x not equal to c.

Now, let's look at a different interval along the function curve. The lowest value of the function in that interval is f(c), and in this case f(c) is the relative minimum or the local minimum.

When x is near c, a function f has a relative minimum at x = c if f(c) is less than f(x) for all x not equal to c.

The relative minimum and maximum are known as relative extrema.

Recall that the slope of a line tangent to the curve is given by the derivative of the curve at that point. Use the slider to vary the x value for the tangent point on the curve. Look at the values displayed for the tangent slope at the relative extrema and nearby values.

What do you observe about the tangent lines at the relative extrema? Click the "Submit" button after selecting your answer.

Fermat's theorem states that if there is a relative maximum or minimum at c, then f′(c) is zero, if it exists. Therefore, a good way to determine extreme values is to look at points where the derivative is zero.

What is the x-value of the relative extremum of this function in the indicated interval? Click the "Submit" button after entering your answer.

Does Fermat's rule imply that if a derivative is equal to zero at c, then c necessarily represents a relative extremum? Before deciding, consider this function.

Is there a relative maximum or minimum at x = 1? Click the "Submit" button after selecting your answer.

A close look at the graph of this function shows that, even though the tangent line at x = 1 is horizontal, no maximum or minimum exists at that point.

Although setting a function's derivative equal to zero is a good way to look for extreme values, the technique actually indicates the existence of critical points, and these points are not necessarily extremum.

Let's look at another function. The function graphed here has a maximum at x = 3.

What is the derivative of the function at x = 3? Click the "Submit" button after selecting your answer.

Another way to express Fermat's theorem is, if a function has a minimum or maximum at c, then c is a critical number.

In general, if the derivative f′(c) equals zero or does not exist, then the number c is known as a critical number.

In this case, the derivative f′(3) does not exist.

The derivative of this function is zero when x equals one.

Copyright 2006 The Regents of the University of California and Monterey Institute for Technology and Education