# Optimizing f(x)

## 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:

1. the point is a (local) maximum;
2. the point is a (local) minimum;
3. 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.

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