Lösung 1.3:2b

Aus Online Mathematik Brückenkurs 2

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

In our case, we have that:

1. The derivative of f(x) is given by

   f(x)=3−2x

   and becomes zero when x=32.
2. The function is a polynomial, and is therefore differentiable everywhere.
3. The function is defined for all x, and therefore the interval of definition has no endpoints.

There is thus just one point x=32, where the function possibly has an extreme point.

If we write down a sign table for the derivative, we see that x=32 is a local maximum.

x		23
f(x)	+	0	−
f(x)		417

Because the function is given by a second-degree expression, its graph is a parabola with a maximum at (32174) and we can draw it with the help of a few couple of points.