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Aus Online Mathematik Brückenkurs 2

As always, a function can only have local extreme points at one of the following types of points:

1. Critical points, i.e. where fx=0 ;

2. Points where the function is not differentiable;

3. Endpoints of the interval of definition.

We investigate these three cases.

1. We obtain the critical points by setting the derivative equal to zero:

fx=x2−x−1ex+x2−x−1ex=2x−1ex+x2−x−1ex=x2+x−2ex

This expression for the derivative can only be zero when x2+x−2=0, because ex differs from zero for all values of x. We solve the second-degree equation by completing the square:

x+212−212−2=0=x+212=49=x+21=23

i.e. x=−21−23=−2 and x=−21+23=1.Both of these points lie within the region of definition, −3x3

2. The function is a polynomial x2−x−1 multiplied by the exponential function ex, and, because both of these functions are differentiable, the product is also a differentiable function, which shows that the function is differentiable everywhere.

3. The function's region of definition is −3x3 and the endpoints x=−3 and x=3 are therefore possible local extreme points.

All in all, there are four points x=−3x=−2x=1 and x=3 where the function possibly has local extreme points.

Now, we will write down a table of the sign of the derivative, in order to investigate the function has local extreme points.

We can factorize the derivative somewhat,

fx=x2+x−2ex=x+2x−1ex , since x2+x−2 has zeros at x=−2 and x=1. Each individual factor in the derivative has a sign that is given in the table:

TABELL

The sign of the derivative is the product of these signs and from the derivative's sign we decide which local extreme points we have:

TABELL

The function has local minimum points at x=−3 and x=1, and local maximum points x=−2 and x=3.