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 \displaystyle {f}'\left( x \right)=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 \displaystyle f\left( x \right) is given by
\displaystyle {f}'\left( x \right)=3-2x
and becomes zero when \displaystyle x=\frac{3}{2}.
2. The function is a polynomial, and is therefore differentiable everywhere.
3. The function is defined for all \displaystyle x, and there are therefore the interval of definition has no endpoints.
There is thus a point \displaystyle x=\frac{3}{2}, where the function possibly has an extreme point.
If we write down a sign table for the derivative, we see that \displaystyle x=\frac{3}{2} is a local maximum.
TABLE
Because the function is given by a second-degree expression, its graph is a parabola with a maximum at \displaystyle \left( \frac{3}{2} \right.,\left. \frac{17}{4} \right) and we can draw it with the help of a few couple of points.
PICTURE TABLE