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Aus Online Mathematik Brückenkurs 2

There are three types of points at which the function can have local extreme points:

1. Critical points, i.e. where fx=0 ;

2. Points where the function is not differentiable;

3. Endpoints of the interval of definition.

Because our function is a polynomial, it is defined and differentiable everywhere, and therefore does not have any points which satisfy 2 and 3.

As regards 1, we set the derivative equal to zero and obtain the equation

fx=6x2+6x−12=0 

Dividing both sides by 6 and completing the square, we obtain

x+212−212−2=0 

This gives us the equation

x+212=49 

and taking the root gives the solutions

x=−21−49=−21−23=−2 

x=−21+49=−21+23=1 

This means that if the function has several extreme points, they must lie between x=−2 and x=1.

Then, we write down a sign table for the derivative, and read off the possible extreme points.

TABLE

The function has a local maximum at x=−2 and a local minimum at x=1.

We obtain the overall appearance of the graph of the function from the table and by calculating the value of the function at a few points.

PICTURE TABLE