# The Sign of the Derivative

**Recall from the previous page:** Let *f* (*x* ) be a function and assume that for each value of *x*. we can calculate the slope of the tangent to the graph *y* = *f* (*x* ) at *x*. This slope depends on the value of *x* that we choose, and so is itself a function. We call this function the derivative of *f* (*x* ) and denote it by *f ´* (*x* ).

## Maximum and Minimum

The graph of a function *y* = *f* (*x* ) has a *local maximum* at the point where the graph changes from increasing to decreasing. At this point the tangent has *zero slope*. The graph has a *local minimum* at the point where the graph changes from decreasing to increasing. Again, at this point the tangent has *zero slope* .

### Exercise

Make sure you understand the following connections between the two graphs.- When the graph of the function
*f*(*x*) has a horizontal tangent then

- When the gradient of the function
*f*(*x*) is positive ,

*f*'(

*x*) is above the

*x* -axis (is positive) .

*f*(

*x*) is negative.

the graph of its derivative *f* '(*x* ) is below the *x* -axis (is negative) .

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This gives a method for finding the minimum or maximum points for a function. See later for the preferred method .

- Differentiate the function,
*f*(*x*). to obtain*f*'(*x*) . - Solve the equation
*f*'(*x*) = 0 for*x*to get the values of*x*at minima or maxima. - For each
*x*value:- Determine the value of
*f*'(*x*) for values a little smaller and a little larger than the*x*value. - Decide whether you have a minimum or a maximum.
- Calculate the value of the function at the
*x*value.

- Determine the value of

### Exercise

To see some worked examples, get a new exercise and immediately click show answer until you are confident.

Source: mathsfirst.massey.ac.nz

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