In he unrection UI a unil vector U= y Du} = V ƒ •u= f,uj + fyu2. By t e theorem, D„(Duf)= V(Duƒ)·u= :(Duf)u2. Continue this calculation to %3D || that D(Duf)= fxxu¡ + 2 fxyuju2 + e the square to get Du(Duf) = fxx (u1 * (fxx fyy – f,). - fxx (why we can find an open ball B centered at ) such that f,(x, y) < 0 and D(x, y) > x, y) in B. Conclude that f (xo, yo) is a (, y) in B. Show that D„(Duƒ)(x, y) < 0 fo kimum. es 3 to 6: Looking at the gradient vector fiel (or say there are none) where f has a local min differentiable function f(x, y), identify , local maximum, or saddle point. 11 apter 4. Scalar and Vector Fields 6. 5. 11 У 1. Exercises 7 to 16: Find all critical points (if any) of a given function f(x, y) and determine wike they are local extreme points or saddle points. 7. f(x, y) = x2 + y² + xy²

In he unrection UI a unil vector U= y Du} = V ƒ •u= f,uj + fyu2. By t e theorem, D„(Duf)= V(Duƒ)·u= :(Duf)u2. Continue this calculation to %3D || that D(Duf)= fxxu¡ + 2 fxyuju2 + e the square to get Du(Duf) = fxx (u1 * (fxx fyy – f,). - fxx (why we can find an open ball B centered at ) such that f,(x, y) < 0 and D(x, y) > x, y) in B. Conclude that f (xo, yo) is a (, y) in B. Show that D„(Duƒ)(x, y) < 0 fo kimum. es 3 to 6: Looking at the gradient vector fiel (or say there are none) where f has a local min differentiable function f(x, y), identify , local maximum, or saddle point. 11 apter 4. Scalar and Vector Fields 6. 5. 11 У 1. Exercises 7 to 16: Find all critical points (if any) of a given function f(x, y) and determine wike they are local extreme points or saddle points. 7. f(x, y) = x2 + y² + xy²

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Explain which points on these graphs are local max/min/ saddle points. Also what is a saddle point? You should be able to tell from looking at the vector field, no equations are given.

In he unrection UI a unil vector U=
y Du} = V ƒ •u= f,uj + fyu2. By t e theorem, D„(Duf)= V(Duƒ)·u=
:(Duf)u2. Continue this calculation to
%3D
||
that D(Duf)= fxxu¡ + 2 fxyuju2 +
e the square to get Du(Duf) = fxx (u1
* (fxx fyy – f,).
-
fxx
(why we can find an open ball B centered at
) such that f,(x, y) < 0 and D(x, y) >
x, y) in B. Conclude that f (xo, yo) is a
(, y) in B. Show that D„(Duƒ)(x, y) < 0 fo
kimum.
es 3 to 6: Looking at the gradient vector fiel
(or say there are none) where f has a local min
differentiable function f(x, y), identify
, local maximum, or saddle point.

11
apter 4. Scalar and Vector Fields
6.
5.
11
У
1.
Exercises 7 to 16: Find all critical points (if any) of a given function f(x, y) and determine wike
they are local extreme points or saddle points.
7. f(x, y) = x2 + y² + xy²

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

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