Maksimum og minimum f(x,y,z)

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Elena Malyutina

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4 years, 1 month ago

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Elena Malyutina 4 years, 1 month ago

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Elena Malyutina 4 years, 1 month ago

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Elena Malyutina 4 years, 1 month ago

Created this as a copy of Maksimum og minimum f(x,y).

Name	Status	Author	Last Modified
Maksimum og minimum f(x,y)	Ready to use	Elena Malyutina	09/02/2022 14:29
Maksimum og minimum f(x,y,z)	Ready to use	Elena Malyutina	10/02/2022 07:44

Question statement

La funksjonen $f:X\subseteq\mathbb{R}^3\rightarrow\mathbb{R}$ være deriverbar i $\mathbf{a}$ .

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La $f_x(\mathbf{a})=f_y(\mathbf{a})=f_z(\mathbf{a})=0$ .

La $d_1=f_{xx}(\mathbf{a})$ , $d_2=\left| \begin{array}{ccc} f_{xx}(\mathbf{a}) & f_{xy}(\mathbf{a}) \\ f_{yx}(\mathbf{a}) & f_{yy}(\mathbf{a}) \end{array} \right|$ , $d_3=\left| \begin{array}{ccc} f_{xx}(\mathbf{a}) & f_{xy}(\mathbf{a}) & f_{xx}(\mathbf{a}) \\ f_{yx}(\mathbf{a}) & f_{yy}(\mathbf{a}) & f_{yz}(\mathbf{a}) \\ f_{zx}(\mathbf{a}) & f_{zy}(\mathbf{a}) & f_{zz}(\mathbf{a})\end{array} \right|$ ,

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$\$
$d_1<0 \wedge d_2>0 \wedge d_3<0$
$\$
$\$
$d_1>0 \wedge d_2>0 \wedge d_3>0$
$\$
$\$
$d_3\neq0$ , men ingen av de to første påstandene er sanne
$\$
$\$
$d_3=0$
$\$

$\mathbf{a}$ er et lokalt maksimum
$\mathbf{a}$ er et lokalt minimum
$\mathbf{a}$ er et sadel punkt
det kan ingenting sies om $\mathbf{a}$

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	$\mathbf{a}$ er et lokalt maksimum	$\mathbf{a}$ er et lokalt minimum	$\mathbf{a}$ er et sadel punkt	det kan ingenting sies om $\mathbf{a}$
$\$ $d_1<0 \wedge d_2>0 \wedge d_3<0$ $\$
$\$ $d_1>0 \wedge d_2>0 \wedge d_3>0$ $\$
$\$ $d_3\neq0$ , men ingen av de to første påstandene er sanne $\$
$\$ $d_3=0$ $\$

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This question is used in the following exams:

Uke 10 by Elena Malyutina in Elena's workspace.
Funksjoner av flere variabler by Elena Malyutina in Elena's workspace.
Numbastest: maksimum, minimum og sadelpunkter by Ida Friestad Pedersen in Elena's workspace.
Uke 17 by Glen Wilson in Glen's workspace.
Ekstremalpunkter by Glen Wilson in Glen's workspace.

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Taxonomy: mathcentre

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Elena Malyutina 4 years, 1 month ago

Elena Malyutina 4 years, 1 month ago

Elena Malyutina 4 years, 1 month ago

Error in variable testing condition

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