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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 f(x)=0,
2. points where the function is not differentiable, and
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 items 2 and 3.

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

f(x)=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 square root gives the solutions

xx=−21−49=−21−23=−2=−21+49=−21+23=1.

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

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

x		−2		1
f(x)	+	0	−	0	+
f(x)		21		−6

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.