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Lösung 1.3:2b

Aus Online Mathematik Brückenkurs 2

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In order to determine the function's extreme points, we investigate three 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.

In our case, we have that:

1. The derivative of fx  is given by

fx=32x 

and becomes zero when x=23.

2. The function is a polynomial, and is therefore differentiable everywhere.

3. The function is defined for all x, and there are therefore the interval of definition has no endpoints.

There is thus a point x=23, where the function possibly has an extreme point.

If we write down a sign table for the derivative, we see that x=23 is a local maximum.

TABLE

Because the function is given by a second-degree expression, its graph is a parabola with a maximum at 23417  and we can draw it with the help of a few couple of points.

PICTURE TABLE