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Lösung 1.3:1c

Aus Online Mathematik Brückenkurs 2

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The function has zero derivative at three points,
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The function has zero derivative at three points, <math>x=a</math>, <math>x=b</math> and <math>x=c</math> (see picture below), which are therefore the critical points of the function.
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<math>x=a</math>,
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<math>x=b</math>and
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<math>x=c</math>
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(see picture below), which are therefore the critical points of the function.
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[[Image:1_3_1_c1.gif|center]]
[[Image:1_3_1_c1.gif|center]]
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The point
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The point <math>x=b</math> is an inflexion point because the derivative is positive in a neighbourhood both to the left and right.
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<math>x=b</math>
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is an inflexion point because the derivative is positive in a neighbourhood both the left and right.
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At the left endpoint of the interval of definition and at
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At the left endpoint of the interval of definition and at <math>x=c</math>, the function has local maximum points , because the function takes lower values at all points in the vicinity of these points. At the point <math>x=a</math> and the right endpoint, the function has local minimum points.
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<math>x=c</math>, the function has local maximum points , because the function takes lower values at all points in the vicinity of these points. At the point
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<math>x=a</math>
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and the right endpoint, the function has local minimum points.
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Also, we see that
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<math>x=c</math>
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is a global maximum (the function takes its largest value there) and the right endpoint is a global minimum.
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Also, we see that <math>x=c</math> is a global maximum (the function takes its largest value there) and the right endpoint is a global minimum.
[[Image:1_3_1_c2.gif|center]]
[[Image:1_3_1_c2.gif|center]]
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Between the left endpoint and
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Between the left endpoint and <math>x=a</math>, as well as between <math>x=c</math>
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<math>x=a</math>, as well as between
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and the right endpoint, the function is strictly decreasing (the larger <math>x</math> is, the smaller <math>f(x)</math> becomes), whilst the function is strictly increasing between <math>x=a</math> and <math>x=c</math> (the graph flattens out at <math>x=b</math>, but it isn't constant there).
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<math>x=c</math>
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and the right endpoint, the function is strictly decreasing (the larger
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<math>x</math>
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is, the smaller
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<math>f\left( x \right)</math>
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becomes), whilst the function is strictly increasing between
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<math>x=a</math>
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and x=c (the graph flattens out at
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<math>x=b</math>, but it isn't constant there).
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[[Image:1_3_1_c3.gif|center]]
[[Image:1_3_1_c3.gif|center]]

Version vom 12:03, 17. Okt. 2008

The function has zero derivative at three points, x=a, x=b and x=c (see picture below), which are therefore the critical points of the function.

The point x=b is an inflexion point because the derivative is positive in a neighbourhood both to the left and right.

At the left endpoint of the interval of definition and at x=c, the function has local maximum points , because the function takes lower values at all points in the vicinity of these points. At the point x=a and the right endpoint, the function has local minimum points.

Also, we see that x=c is a global maximum (the function takes its largest value there) and the right endpoint is a global minimum.

Between the left endpoint and x=a, as well as between x=c and the right endpoint, the function is strictly decreasing (the larger x is, the smaller f(x) becomes), whilst the function is strictly increasing between x=a and x=c (the graph flattens out at x=b, but it isn't constant there).

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