Step 1The function z =f(x, y) describes a surface in three dimensions. Therefore, for a maximum or minimum pointto exist, a horizontal plane must be tangent to the surface at that point. For a plane to be horizontal, everyline on the plane will also be horizontal. Therefore, tangent lines parallel to the x- and y-axes will have zeroslopes. Thus, to find any critical points, we need to determine where both Zx and zy are equal to zero.Begin by finding the partial derivatives of z with respect to x and y.|-2x + yZxу — 2гZy = x – 2y + 212y + 21Step 2Next, we find any points that satisfy both Zx= 0 and zy = 0. Setting zx equal to zero allows us to easily%3Dexpress y in terms of x.— 2х + у %3D 02xy =2.xStep 3Now, we can set zy = 0, substitute 2x forY,and solve for x.х — 2+ 21 = 0= -21X =

Substitution method: - The substitution technique could be a simple approach for determining linear…

Answered: Now, we can set zy = 0, substitute 2x…

Math

Calculus

Now, we can set zy = 0, substitute 2x for y, and solve for x. + 21 = 0 = -21 X =

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter3: Functions

Section3.3: More On Functions; Piecewise-defined Functions

Problem 99E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...

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Question

Step 1
The function z =
f(x, y) describes a surface in three dimensions. Therefore, for a maximum or minimum point
to exist, a horizontal plane must be tangent to the surface at that point. For a plane to be horizontal, every
line on the plane will also be horizontal. Therefore, tangent lines parallel to the x- and y-axes will have zero
slopes. Thus, to find any critical points, we need to determine where both Zx and zy are equal to zero.
Begin by finding the partial derivatives of z with respect to x and y.
|-2x + y
Zx
у — 2г
Zy = x – 2y + 21
2y + 21
Step 2
Next, we find any points that satisfy both Zx
= 0 and zy = 0. Setting zx equal to zero allows us to easily
%3D
express y in terms of x.
— 2х + у %3D 0
2x
y =
2.x
Step 3
Now, we can set zy = 0, substitute 2x for
Y,
and solve for x.
х — 2
+ 21 = 0
= -21
X =

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