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Find the function's relative maxima, relative minima, and saddle points, if they exist. z = 19 x² + xy - y² + 21y 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. y-2r Fantastic job! Step 2 Zx=y-2x Zy= - 2y + 21 Step 4 Next, we find any points that satisfy both zx = 0 and 2y = 0. Setting Zx equal to zero allows us to easily express y in terms of x. -2x + y = 0 x-2y+21 Great job! y Step 3 Now, we can set zy= 0, substitute 2x for y, and solve for x. 2x 2x 2x x Impressive work! +21=0 -3.x -21 x=7✔ 7 Nicely done.

Find the function's relative maxima, relative minima, and saddle points, if they exist. z = 19 x² + xy - y² + 21y 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. y-2r Fantastic job! Step 2 Zx=y-2x Zy= - 2y + 21 Step 4 Next, we find any points that satisfy both zx = 0 and 2y = 0. Setting Zx equal to zero allows us to easily express y in terms of x. -2x + y = 0 x-2y+21 Great job! y Step 3 Now, we can set zy= 0, substitute 2x for y, and solve for x. 2x 2x 2x x Impressive work! +21=0 -3.x -21 x=7✔ 7 Nicely done.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Find the function's relative maxima, relative minima, and saddle points, if they exist. z = 19 x² + xy - y² + 21y 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. y-2r Fantastic job! Zx = Zy = 2x - 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 express y in terms of x. -2x + y = 0 y = 2x x-2y+21 Great job! (x, y, z) = Step 3 Now, we can set zy = 0, substitute 2x for y, and solve for x. 2x 2x X 2x Impressive work! X + 21 = 0 -3x -21 x=7 ✔ Step 4 Use the value of x to solve for y. Then use those values in the original equation to solve for z and determine the critical point. 7 Nicely done.
Transcribed Image Text:Find the function's relative maxima, relative minima, and saddle points, if they exist. z = 19 x² + xy - y² + 21y 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. y-2r Fantastic job! Zx = Zy = 2x - 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 express y in terms of x. -2x + y = 0 y = 2x x-2y+21 Great job! (x, y, z) = Step 3 Now, we can set zy = 0, substitute 2x for y, and solve for x. 2x 2x X 2x Impressive work! X + 21 = 0 -3x -21 x=7 ✔ Step 4 Use the value of x to solve for y. Then use those values in the original equation to solve for z and determine the critical point. 7 Nicely done.
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