## Summary and examples

is the derivative of a function in a point and is the slope of the tangent line at that point of the graph of that function.

When it means that the tangent line in has a slope and thus that it is horizontal to the -axis. For such a point there are three possibilities:

- the point is a (local) maximum;
- the point is a (local) minimum;
- the point is a horizontal inflection point (an example is the graph of ).

When we want to know in which points a function has a local maximum, minimum or horizontal inflection point, we calculate the derivative and try to find for which values , for these values of the derivative equals zero and the tangent line is horizontal.

##### Example 1

Determine the minimum of the parabola:

We already know that this parabola ‘opens up’ (the coefficient of the square equals and thus ) and thus has a minimum that lies on the symmetry-axis, see Quadratic functions and graphs.

We differentiate the function and get:

and thus:

The parabola has a minimum in the point .

##### Example 2

Quadratic functions (the graphs are parabolas):

have a maximum or minimum that lies on the symmetry-axis:

We can also calculate this by differentiation. The derivative of the function above is:

The derivative is equal to when:

which is the -coordinate of the top.

The values of for which the derivative of the function equals are candidates for a maximum, minimum or horizontal inflection point, but we do not know yet which of the three is the case. This requires further investigation. There are two different methods for this.

*Method 1: consider the sign of the first derivative*

The derivative of a maximum in a point is equal to .

Left to the maximum the function is increasing and thus for .

Right to the maximum the function is decreasing and thus for .

So the sign of the first derivative around the maximum has the following signs:

*maximum *: + -.

Similarly, the sign of the first derivative around the minimum has the following signs:

*minimum*: – +.

For a horizontal inflection point we find similarly:

*horizontal inflection point *: + + or – -.

*Method 2: consider the 2nd derivative*

When is one of the solutions of the equation it is a candidate for a maximum, minimum or horizontal inflection point. Then the following holds.

If:

, then is a minimum;

, then is a maximum;

, then is a horizontal inflection point.

##### Example 3

Check whether the function:

has a maximum and/or a minimum.

The derivative is:

The derivative equals if:

so if:

or

The corresponding points on the graph are, respectively:

or

Now we check both methods.

*Method 1*

Around we find the following signs of :

+ – and thus there is a maximum.

Around we find the following signs of :

– + and thus there is a minimum.

*Method 2*

We have to calculate the second derivative:

Then:

and thus the function has a maximum for

and

and thus the function has a minimum for

##### Example 4

Check whether the function:

has a local maximum or minimum.

The derivative of the function is, using the product rule:

The derivative is when and thus the point is a candidate for either a maximum or minimum. Further research should reveal which of the two it is. It is even possible that it is a point of inflection.

We can use either method, but the second method is more laborious because then we need to use the production rule again. Therefore we take method 1 and only need to consider the sign of because the exponential function is always positive.

If then is negative and if then is positief and thus we have a minimum, see the figure.

##### Example 5

Investigate the function:

We calculate the derivative:

The equation:

has as a solution which is a candidate for a maximum, minimum of horizontal inflection point.

The second derivative is:

which is equal for . So the point is a horizontal inflection point.