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Lösung 1.3:1b

Aus Online Mathematik Brückenkurs 2

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There are two points,
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There are two points, <math>x=a</math> and <math>x=b</math> (see picture below), where the function has a horizontal tangent and hence a derivative equal to zero. These are the functions critical points.
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<math>x=a</math>
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and
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<math>x=b</math>
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(see picture below), where the function has a horizontal tangent and hence a derivative equal to zero. These are the functions critical points.
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[[Image:1_3_1_b1.gif|center]]
[[Image:1_3_1_b1.gif|center]]
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Furthermore, we see that the function has local minimum points at the left endpoint of the interval of definition and
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Furthermore, we see that the function has local minimum points at the left endpoint of the interval of definition and <math>x=b</math>, because the function takes higher values at neighbouring points. In the same way, we see that the function has local maximum points at <math>x=a</math> and at the right endpoint of the interval of definition.
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<math>x=b</math>, because the function takes higher values at neighbouring points. In the same way, we see that the function has local maximum points at
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<math>x=a</math>
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and at the right endpoint of the interval of definition.
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Of these local extreme points, the left endpoint is a global minimum (that point where the function takes its absolute minimum value) and the right endpoint is a global maximum.
Of these local extreme points, the left endpoint is a global minimum (that point where the function takes its absolute minimum value) and the right endpoint is a global maximum.
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[[Image:1_3_1_b2.gif|center]]
[[Image:1_3_1_b2.gif|center]]
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The function is strictly increasing (it has a tangent that slopes upwards) in the interval between the left endpoint and
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The function is strictly increasing (it has a tangent that slopes upwards) in the interval between the left endpoint and <math>x=a</math>, as well as between <math>x=b</math> and the right endpoint. In the interval between <math>x=0</math>
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<math>x=a</math>, as well as between
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and <math>x=b</math>, the function is strictly decreasing.
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<math>x=b</math>
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and the right endpoint. In the interval between
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<math>x=0</math>
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and
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<math>x=b</math>, the function is strictly decreasing.
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[[Image:1_3_1_b3.gif|center]]
[[Image:1_3_1_b3.gif|center]]

Version vom 11:54, 17. Okt. 2008

There are two points, x=a and x=b (see picture below), where the function has a horizontal tangent and hence a derivative equal to zero. These are the functions critical points.

Furthermore, we see that the function has local minimum points at the left endpoint of the interval of definition and x=b, because the function takes higher values at neighbouring points. In the same way, we see that the function has local maximum points at x=a and at the right endpoint of the interval of definition.

Of these local extreme points, the left endpoint is a global minimum (that point where the function takes its absolute minimum value) and the right endpoint is a global maximum.

The function is strictly increasing (it has a tangent that slopes upwards) in the interval between the left endpoint and x=a, as well as between x=b and the right endpoint. In the interval between x=0 and x=b, the function is strictly decreasing.

Von „http://wiki.math.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/L%C3%B6sung_1.3:1b“