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Find the absolute maximum and minimum of f(x, y) = 4x2 + 2xy - 3y2 on the unit square0 ≤\le x ≤\le 1, 0 ≤\le y ≤\le 1.


A) maximum Find the absolute maximum and minimum of f(x, y) = 4x<sup>2</sup> + 2xy - 3y<sup>2</sup> on the unit square0 \le x \le 1, 0 \le y \le 1. A) maximum , minimum -3 B) maximum 4, minimum -3 C) maximum 4, minimum 0 D) maximum , minimum -4 E) maximum , minimum 0 , minimum -3
B) maximum 4, minimum -3
C) maximum 4, minimum 0
D) maximum Find the absolute maximum and minimum of f(x, y) = 4x<sup>2</sup> + 2xy - 3y<sup>2</sup> on the unit square0 \le x \le 1, 0 \le y \le 1. A) maximum , minimum -3 B) maximum 4, minimum -3 C) maximum 4, minimum 0 D) maximum , minimum -4 E) maximum , minimum 0 , minimum -4
E) maximum Find the absolute maximum and minimum of f(x, y) = 4x<sup>2</sup> + 2xy - 3y<sup>2</sup> on the unit square0 \le x \le 1, 0 \le y \le 1. A) maximum , minimum -3 B) maximum 4, minimum -3 C) maximum 4, minimum 0 D) maximum , minimum -4 E) maximum , minimum 0 , minimum 0

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Find the absolute maximum and minimum values of f(x, y) = 4(x - x2) sin( π\pi y) on the rectangle 0 ≤\le x ≤\le 1, 0 ≤\le y ≤\le 2 and the points where they are assumed.


A) maximum 1 at Find the absolute maximum and minimum values of f(x, y) = 4(x - x<sup>2</sup>) sin( \pi y) on the rectangle 0 \le x \le 1, 0 \le y \le 2 and the points where they are assumed. A) maximum 1 at , minimum 0 at B) maximum 2 at , minimum -2 at C) maximum 1 at , minimum -1 at D) maximum 1 at , minimum 0 at (0, 0) , (1, 0) , (0, 2) , and (1, 2) E) maximum 1 at , minimum 0 at , minimum 0 at Find the absolute maximum and minimum values of f(x, y) = 4(x - x<sup>2</sup>) sin( \pi y) on the rectangle 0 \le x \le 1, 0 \le y \le 2 and the points where they are assumed. A) maximum 1 at , minimum 0 at B) maximum 2 at , minimum -2 at C) maximum 1 at , minimum -1 at D) maximum 1 at , minimum 0 at (0, 0) , (1, 0) , (0, 2) , and (1, 2) E) maximum 1 at , minimum 0 at
B) maximum 2 at Find the absolute maximum and minimum values of f(x, y) = 4(x - x<sup>2</sup>) sin( \pi y) on the rectangle 0 \le x \le 1, 0 \le y \le 2 and the points where they are assumed. A) maximum 1 at , minimum 0 at B) maximum 2 at , minimum -2 at C) maximum 1 at , minimum -1 at D) maximum 1 at , minimum 0 at (0, 0) , (1, 0) , (0, 2) , and (1, 2) E) maximum 1 at , minimum 0 at , minimum -2 at Find the absolute maximum and minimum values of f(x, y) = 4(x - x<sup>2</sup>) sin( \pi y) on the rectangle 0 \le x \le 1, 0 \le y \le 2 and the points where they are assumed. A) maximum 1 at , minimum 0 at B) maximum 2 at , minimum -2 at C) maximum 1 at , minimum -1 at D) maximum 1 at , minimum 0 at (0, 0) , (1, 0) , (0, 2) , and (1, 2) E) maximum 1 at , minimum 0 at
C) maximum 1 at Find the absolute maximum and minimum values of f(x, y) = 4(x - x<sup>2</sup>) sin( \pi y) on the rectangle 0 \le x \le 1, 0 \le y \le 2 and the points where they are assumed. A) maximum 1 at , minimum 0 at B) maximum 2 at , minimum -2 at C) maximum 1 at , minimum -1 at D) maximum 1 at , minimum 0 at (0, 0) , (1, 0) , (0, 2) , and (1, 2) E) maximum 1 at , minimum 0 at , minimum -1 at Find the absolute maximum and minimum values of f(x, y) = 4(x - x<sup>2</sup>) sin( \pi y) on the rectangle 0 \le x \le 1, 0 \le y \le 2 and the points where they are assumed. A) maximum 1 at , minimum 0 at B) maximum 2 at , minimum -2 at C) maximum 1 at , minimum -1 at D) maximum 1 at , minimum 0 at (0, 0) , (1, 0) , (0, 2) , and (1, 2) E) maximum 1 at , minimum 0 at
D) maximum 1 at Find the absolute maximum and minimum values of f(x, y) = 4(x - x<sup>2</sup>) sin( \pi y) on the rectangle 0 \le x \le 1, 0 \le y \le 2 and the points where they are assumed. A) maximum 1 at , minimum 0 at B) maximum 2 at , minimum -2 at C) maximum 1 at , minimum -1 at D) maximum 1 at , minimum 0 at (0, 0) , (1, 0) , (0, 2) , and (1, 2) E) maximum 1 at , minimum 0 at , minimum 0 at (0, 0) , (1, 0) , (0, 2) , and (1, 2)
E) maximum 1 at Find the absolute maximum and minimum values of f(x, y) = 4(x - x<sup>2</sup>) sin( \pi y) on the rectangle 0 \le x \le 1, 0 \le y \le 2 and the points where they are assumed. A) maximum 1 at , minimum 0 at B) maximum 2 at , minimum -2 at C) maximum 1 at , minimum -1 at D) maximum 1 at , minimum 0 at (0, 0) , (1, 0) , (0, 2) , and (1, 2) E) maximum 1 at , minimum 0 at , minimum 0 at Find the absolute maximum and minimum values of f(x, y) = 4(x - x<sup>2</sup>) sin( \pi y) on the rectangle 0 \le x \le 1, 0 \le y \le 2 and the points where they are assumed. A) maximum 1 at , minimum 0 at B) maximum 2 at , minimum -2 at C) maximum 1 at , minimum -1 at D) maximum 1 at , minimum 0 at (0, 0) , (1, 0) , (0, 2) , and (1, 2) E) maximum 1 at , minimum 0 at

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Suppose that a function f(x,y) has a critical point (a, b) at an interior point in its domain and that f has continuous second order partials in a neighbourhood of (a, b). If Suppose that a function f(x,y) has a critical point (a, b) at an interior point in its domain and that f has continuous second order partials in a neighbourhood of (a, b). If , then f has no local extremum at (a, b). , then f has no local extremum at (a, b).

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If a function f(x,y) has a local or absolute extreme value at the point (x0, y0) in its domain, then (x0, y0) must be either a critical point of f, a singular point of f, or a boundary point of the domain of f.

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Find the absolute maximum and minimum values of the linear function f(x, y) = -2x + y - 10 on the polygon 0 ≤\le x ≤\le 2, 0 ≤\le y ≤\le 2, y - x ≤\le 1.


A) maximum -9, minimum -14
B) maximum -9, minimum -12
C) maximum -8, minimum -12
D) maximum -8, minimum -15
E) maximum -9, minimum -10

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Use Maple's fsolve routine to solve the non-linear system of equations Use Maple's fsolve routine to solve the non-linear system of equations Quote the solution to 5 significant figures. Quote the solution to 5 significant figures.

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(x,y,z) = (0.120759,...

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Let f(x, y, z) = x2 + 2y2 + 4z2. Find the point on the plane x + y + z = 14 at which f has its smallest value.


A) (8, 4, 2)
B) (0, 0, 14)
C) (1, 2, 11)
D) Let f(x, y, z) = x<sup>2</sup> + 2y<sup>2</sup> + 4z<sup>2</sup>. Find the point on the plane x + y + z = 14 at which f has its smallest value. A) (8, 4, 2) B) (0, 0, 14) C) (1, 2, 11) D) E) (2, 4, 8)
E) (2, 4, 8)

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Use Lagrange multipliers to find the maximum and minimum values of the functionf(x, y, z) = xy2z3 on the sphere x2 + y2 + z2 = 6.


A) maximum 6, minimum -6
B) maximum 6 Use Lagrange multipliers to find the maximum and minimum values of the functionf(x, y, z) = xy<sup>2</sup>z<sup>3</sup> on the sphere x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> = 6. A) maximum 6, minimum -6 B) maximum 6 , minimum -6 C) maximum 6 , minimum -6 D) maximum 12, minimum -12 E) maximum 12 , minimum -12 , minimum -6 Use Lagrange multipliers to find the maximum and minimum values of the functionf(x, y, z) = xy<sup>2</sup>z<sup>3</sup> on the sphere x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> = 6. A) maximum 6, minimum -6 B) maximum 6 , minimum -6 C) maximum 6 , minimum -6 D) maximum 12, minimum -12 E) maximum 12 , minimum -12
C) maximum 6 Use Lagrange multipliers to find the maximum and minimum values of the functionf(x, y, z) = xy<sup>2</sup>z<sup>3</sup> on the sphere x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> = 6. A) maximum 6, minimum -6 B) maximum 6 , minimum -6 C) maximum 6 , minimum -6 D) maximum 12, minimum -12 E) maximum 12 , minimum -12 , minimum -6 Use Lagrange multipliers to find the maximum and minimum values of the functionf(x, y, z) = xy<sup>2</sup>z<sup>3</sup> on the sphere x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> = 6. A) maximum 6, minimum -6 B) maximum 6 , minimum -6 C) maximum 6 , minimum -6 D) maximum 12, minimum -12 E) maximum 12 , minimum -12
D) maximum 12, minimum -12
E) maximum 12 Use Lagrange multipliers to find the maximum and minimum values of the functionf(x, y, z) = xy<sup>2</sup>z<sup>3</sup> on the sphere x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> = 6. A) maximum 6, minimum -6 B) maximum 6 , minimum -6 C) maximum 6 , minimum -6 D) maximum 12, minimum -12 E) maximum 12 , minimum -12 , minimum -12 Use Lagrange multipliers to find the maximum and minimum values of the functionf(x, y, z) = xy<sup>2</sup>z<sup>3</sup> on the sphere x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> = 6. A) maximum 6, minimum -6 B) maximum 6 , minimum -6 C) maximum 6 , minimum -6 D) maximum 12, minimum -12 E) maximum 12 , minimum -12

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Find the derivative of the function f(x) = Find the derivative of the function f(x) = dt. A) 6x B) 4x C) 2x D) 2x E) 2x dt.


A) 6x Find the derivative of the function f(x) = dt. A) 6x B) 4x C) 2x D) 2x E) 2x
B) 4x Find the derivative of the function f(x) = dt. A) 6x B) 4x C) 2x D) 2x E) 2x
C) 2x Find the derivative of the function f(x) = dt. A) 6x B) 4x C) 2x D) 2x E) 2x
D) 2x Find the derivative of the function f(x) = dt. A) 6x B) 4x C) 2x D) 2x E) 2x
E) 2x Find the derivative of the function f(x) = dt. A) 6x B) 4x C) 2x D) 2x E) 2x

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Find and classify the critical points of the following function: f(x, y) = Find and classify the critical points of the following function: f(x, y) = + 30x<sup>3</sup> - 15y<sup>3</sup>. A) saddle points are ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) B) saddle points are (0, 0) , ( , - ) and (- , ) , minimum at (3, 3) and maximum at (-3, -3) C) saddle points are (0, 0) , ( , - ) , (- , ) , (3, 3) , and (-3, -3) D) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) E) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, -3) and minimum at (-3, 3) + 30x3 - 15y3.


A) saddle points are ( Find and classify the critical points of the following function: f(x, y) = + 30x<sup>3</sup> - 15y<sup>3</sup>. A) saddle points are ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) B) saddle points are (0, 0) , ( , - ) and (- , ) , minimum at (3, 3) and maximum at (-3, -3) C) saddle points are (0, 0) , ( , - ) , (- , ) , (3, 3) , and (-3, -3) D) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) E) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, -3) and minimum at (-3, 3) , - Find and classify the critical points of the following function: f(x, y) = + 30x<sup>3</sup> - 15y<sup>3</sup>. A) saddle points are ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) B) saddle points are (0, 0) , ( , - ) and (- , ) , minimum at (3, 3) and maximum at (-3, -3) C) saddle points are (0, 0) , ( , - ) , (- , ) , (3, 3) , and (-3, -3) D) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) E) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, -3) and minimum at (-3, 3) ) and (- Find and classify the critical points of the following function: f(x, y) = + 30x<sup>3</sup> - 15y<sup>3</sup>. A) saddle points are ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) B) saddle points are (0, 0) , ( , - ) and (- , ) , minimum at (3, 3) and maximum at (-3, -3) C) saddle points are (0, 0) , ( , - ) , (- , ) , (3, 3) , and (-3, -3) D) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) E) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, -3) and minimum at (-3, 3) , Find and classify the critical points of the following function: f(x, y) = + 30x<sup>3</sup> - 15y<sup>3</sup>. A) saddle points are ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) B) saddle points are (0, 0) , ( , - ) and (- , ) , minimum at (3, 3) and maximum at (-3, -3) C) saddle points are (0, 0) , ( , - ) , (- , ) , (3, 3) , and (-3, -3) D) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) E) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, -3) and minimum at (-3, 3) ) , maximum at (3, 3) and minimum at (-3, -3)
B) saddle points are (0, 0) , ( Find and classify the critical points of the following function: f(x, y) = + 30x<sup>3</sup> - 15y<sup>3</sup>. A) saddle points are ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) B) saddle points are (0, 0) , ( , - ) and (- , ) , minimum at (3, 3) and maximum at (-3, -3) C) saddle points are (0, 0) , ( , - ) , (- , ) , (3, 3) , and (-3, -3) D) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) E) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, -3) and minimum at (-3, 3) , - Find and classify the critical points of the following function: f(x, y) = + 30x<sup>3</sup> - 15y<sup>3</sup>. A) saddle points are ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) B) saddle points are (0, 0) , ( , - ) and (- , ) , minimum at (3, 3) and maximum at (-3, -3) C) saddle points are (0, 0) , ( , - ) , (- , ) , (3, 3) , and (-3, -3) D) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) E) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, -3) and minimum at (-3, 3) ) and (- Find and classify the critical points of the following function: f(x, y) = + 30x<sup>3</sup> - 15y<sup>3</sup>. A) saddle points are ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) B) saddle points are (0, 0) , ( , - ) and (- , ) , minimum at (3, 3) and maximum at (-3, -3) C) saddle points are (0, 0) , ( , - ) , (- , ) , (3, 3) , and (-3, -3) D) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) E) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, -3) and minimum at (-3, 3) , Find and classify the critical points of the following function: f(x, y) = + 30x<sup>3</sup> - 15y<sup>3</sup>. A) saddle points are ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) B) saddle points are (0, 0) , ( , - ) and (- , ) , minimum at (3, 3) and maximum at (-3, -3) C) saddle points are (0, 0) , ( , - ) , (- , ) , (3, 3) , and (-3, -3) D) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) E) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, -3) and minimum at (-3, 3) ) , minimum at (3, 3) and maximum at (-3, -3)
C) saddle points are (0, 0) , ( Find and classify the critical points of the following function: f(x, y) = + 30x<sup>3</sup> - 15y<sup>3</sup>. A) saddle points are ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) B) saddle points are (0, 0) , ( , - ) and (- , ) , minimum at (3, 3) and maximum at (-3, -3) C) saddle points are (0, 0) , ( , - ) , (- , ) , (3, 3) , and (-3, -3) D) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) E) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, -3) and minimum at (-3, 3) , - Find and classify the critical points of the following function: f(x, y) = + 30x<sup>3</sup> - 15y<sup>3</sup>. A) saddle points are ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) B) saddle points are (0, 0) , ( , - ) and (- , ) , minimum at (3, 3) and maximum at (-3, -3) C) saddle points are (0, 0) , ( , - ) , (- , ) , (3, 3) , and (-3, -3) D) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) E) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, -3) and minimum at (-3, 3) ) , (- Find and classify the critical points of the following function: f(x, y) = + 30x<sup>3</sup> - 15y<sup>3</sup>. A) saddle points are ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) B) saddle points are (0, 0) , ( , - ) and (- , ) , minimum at (3, 3) and maximum at (-3, -3) C) saddle points are (0, 0) , ( , - ) , (- , ) , (3, 3) , and (-3, -3) D) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) E) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, -3) and minimum at (-3, 3) , Find and classify the critical points of the following function: f(x, y) = + 30x<sup>3</sup> - 15y<sup>3</sup>. A) saddle points are ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) B) saddle points are (0, 0) , ( , - ) and (- , ) , minimum at (3, 3) and maximum at (-3, -3) C) saddle points are (0, 0) , ( , - ) , (- , ) , (3, 3) , and (-3, -3) D) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) E) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, -3) and minimum at (-3, 3) ) , (3, 3) , and (-3, -3)
D) saddle points are (0, 0) , ( Find and classify the critical points of the following function: f(x, y) = + 30x<sup>3</sup> - 15y<sup>3</sup>. A) saddle points are ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) B) saddle points are (0, 0) , ( , - ) and (- , ) , minimum at (3, 3) and maximum at (-3, -3) C) saddle points are (0, 0) , ( , - ) , (- , ) , (3, 3) , and (-3, -3) D) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) E) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, -3) and minimum at (-3, 3) , - Find and classify the critical points of the following function: f(x, y) = + 30x<sup>3</sup> - 15y<sup>3</sup>. A) saddle points are ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) B) saddle points are (0, 0) , ( , - ) and (- , ) , minimum at (3, 3) and maximum at (-3, -3) C) saddle points are (0, 0) , ( , - ) , (- , ) , (3, 3) , and (-3, -3) D) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) E) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, -3) and minimum at (-3, 3) ) and (- Find and classify the critical points of the following function: f(x, y) = + 30x<sup>3</sup> - 15y<sup>3</sup>. A) saddle points are ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) B) saddle points are (0, 0) , ( , - ) and (- , ) , minimum at (3, 3) and maximum at (-3, -3) C) saddle points are (0, 0) , ( , - ) , (- , ) , (3, 3) , and (-3, -3) D) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) E) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, -3) and minimum at (-3, 3) , Find and classify the critical points of the following function: f(x, y) = + 30x<sup>3</sup> - 15y<sup>3</sup>. A) saddle points are ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) B) saddle points are (0, 0) , ( , - ) and (- , ) , minimum at (3, 3) and maximum at (-3, -3) C) saddle points are (0, 0) , ( , - ) , (- , ) , (3, 3) , and (-3, -3) D) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) E) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, -3) and minimum at (-3, 3) ) , maximum at (3, 3) and minimum at (-3, -3)
E) saddle points are (0, 0) , ( Find and classify the critical points of the following function: f(x, y) = + 30x<sup>3</sup> - 15y<sup>3</sup>. A) saddle points are ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) B) saddle points are (0, 0) , ( , - ) and (- , ) , minimum at (3, 3) and maximum at (-3, -3) C) saddle points are (0, 0) , ( , - ) , (- , ) , (3, 3) , and (-3, -3) D) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) E) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, -3) and minimum at (-3, 3) , - Find and classify the critical points of the following function: f(x, y) = + 30x<sup>3</sup> - 15y<sup>3</sup>. A) saddle points are ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) B) saddle points are (0, 0) , ( , - ) and (- , ) , minimum at (3, 3) and maximum at (-3, -3) C) saddle points are (0, 0) , ( , - ) , (- , ) , (3, 3) , and (-3, -3) D) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) E) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, -3) and minimum at (-3, 3) ) and (- Find and classify the critical points of the following function: f(x, y) = + 30x<sup>3</sup> - 15y<sup>3</sup>. A) saddle points are ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) B) saddle points are (0, 0) , ( , - ) and (- , ) , minimum at (3, 3) and maximum at (-3, -3) C) saddle points are (0, 0) , ( , - ) , (- , ) , (3, 3) , and (-3, -3) D) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) E) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, -3) and minimum at (-3, 3) , Find and classify the critical points of the following function: f(x, y) = + 30x<sup>3</sup> - 15y<sup>3</sup>. A) saddle points are ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) B) saddle points are (0, 0) , ( , - ) and (- , ) , minimum at (3, 3) and maximum at (-3, -3) C) saddle points are (0, 0) , ( , - ) , (- , ) , (3, 3) , and (-3, -3) D) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, 3) and minimum at (-3, -3) E) saddle points are (0, 0) , ( , - ) and (- , ) , maximum at (3, -3) and minimum at (-3, 3) ) , maximum at (3, -3) and minimum at (-3, 3)

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By first differentiating the integral, evaluate By first differentiating the integral, evaluate dy for x > -1. A) ln (2x + 1) B) ln (|x|) + 1 C) ln (x - 1) D) ln (x + 1) E) ln (x) + 1 dy for x > -1.


A) ln (2x + 1)
B) ln (|x|) + 1
C) ln (x - 1)
D) ln (x + 1)
E) ln (x) + 1

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If the Lagrange function L corresponding to the problem of extremizing f(x, y, z) subject to the constraint g(x, y, z) = 0 has exactly two critical points, then f must attain its maximum value at one of the points and attain its minimum value at the other point.

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Find the maximum and minimum distances from the origin to the ellipse 5x2 + 6xy + 5y2 - 8 = 0.


A) maximum 2, minimum 1
B) maximum 4, minimum 1
C) maximum 2, minimum Find the maximum and minimum distances from the origin to the ellipse 5x<sup>2</sup> + 6xy + 5y<sup>2</sup> - 8 = 0. A) maximum 2, minimum 1 B) maximum 4, minimum 1 C) maximum 2, minimum D) maximum 4, minimum E) maximum 4, minimum 2
D) maximum 4, minimum Find the maximum and minimum distances from the origin to the ellipse 5x<sup>2</sup> + 6xy + 5y<sup>2</sup> - 8 = 0. A) maximum 2, minimum 1 B) maximum 4, minimum 1 C) maximum 2, minimum D) maximum 4, minimum E) maximum 4, minimum 2
E) maximum 4, minimum 2

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The extreme values of the function f(x , y, z) = 23 x + y2z subject to the constraintsx - z = 0 and y2 + z2 = 36 are given by:


A) 115, 128
B) - 115, 115
C) -128, 115
D) -128, 128
E) -128, -115

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Find and classify all critical points of f(x,y,z) = x3 + xz2 + 3x2 + y2 + 2z2 - 9x - 2y -10.


A) local minimum at (-2, 1, 3) , (-2, 1, -3) , (-3, 1, 0) and saddle point at (1, 1, 0)
B) local minimum at (-2, 1, 3) , (-2, 1, -3) and local maximum at (-3, 1, 0) , (1, 1, 0)
C) local minimum at (1, 1, 0 ) and saddle point at (-2, 1, 3) , (-2, 1, -3) , (-3, 1, 0)
D) local maximum at (-2, 1, 3) , (-2, 1, -3) , (-3, 1, 0) and saddle point at (1, 1, 0)
E) local minimum at (-2, 1, 3) , (-3, 1, 0) , local maximum at (1, 1, 0) , and saddle point at (-2, 1, -3)

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Find and classify all critical points for the function f(x, y) = x3 - 12xy2 + y3 + 45y.


A) (2, 1) and (-2, -1) are saddle points
B) (2, 1) , (2, -1) , (-2, 1) , and (-2, -1) are saddle points
C) (2, 1) is a local maximum and (-2, -1) is a local minimum
D) (2, -1) and (-2, 1) are saddle points
E) There are no critical points.

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Find the point on the sphere x2 + y2 + z2 = 10 that is closest to the point (1, -8, 5) .


A) Find the point on the sphere x<sup>2 </sup>+ y<sup>2 </sup>+ z<sup>2</sup> = 10 that is closest to the point (1, -8, 5) . A) B) C) (0, , 0) D) E) (2, -16, 10)
B) Find the point on the sphere x<sup>2 </sup>+ y<sup>2 </sup>+ z<sup>2</sup> = 10 that is closest to the point (1, -8, 5) . A) B) C) (0, , 0) D) E) (2, -16, 10)
C) (0, Find the point on the sphere x<sup>2 </sup>+ y<sup>2 </sup>+ z<sup>2</sup> = 10 that is closest to the point (1, -8, 5) . A) B) C) (0, , 0) D) E) (2, -16, 10) , 0)
D) Find the point on the sphere x<sup>2 </sup>+ y<sup>2 </sup>+ z<sup>2</sup> = 10 that is closest to the point (1, -8, 5) . A) B) C) (0, , 0) D) E) (2, -16, 10)
E) (2, -16, 10)

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Find and classify the critical points of the Lagrange function L( Find and classify the critical points of the Lagrange function L( , ,..., , λ) corresponding to the problem:extremize subject to = . , Find and classify the critical points of the Lagrange function L( , ,..., , λ) corresponding to the problem:extremize subject to = . ,..., Find and classify the critical points of the Lagrange function L( , ,..., , λ) corresponding to the problem:extremize subject to = . , λ) corresponding to the problem:extremize Find and classify the critical points of the Lagrange function L( , ,..., , λ) corresponding to the problem:extremize subject to = . subject to = Find and classify the critical points of the Lagrange function L( , ,..., , λ) corresponding to the problem:extremize subject to = . . Find and classify the critical points of the Lagrange function L( , ,..., , λ) corresponding to the problem:extremize subject to = . Find and classify the critical points of the Lagrange function L( , ,..., , λ) corresponding to the problem:extremize subject to = .

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Critical point at ( blured image , blured image ,...,

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Find the point on the surface z = x2 + y2 closest to the point (1, 1, 0) .


A) (1, 1, 1)
B) Find the point on the surface z = x<sup>2</sup> + y<sup>2</sup> closest to the point (1, 1, 0) . A) (1, 1, 1) B) C) D) E)
C) Find the point on the surface z = x<sup>2</sup> + y<sup>2</sup> closest to the point (1, 1, 0) . A) (1, 1, 1) B) C) D) E)
D) Find the point on the surface z = x<sup>2</sup> + y<sup>2</sup> closest to the point (1, 1, 0) . A) (1, 1, 1) B) C) D) E)
E) Find the point on the surface z = x<sup>2</sup> + y<sup>2</sup> closest to the point (1, 1, 0) . A) (1, 1, 1) B) C) D) E)

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Let pi > 0, i = 1, 2, 3,..., n be real numbers such that Let pi > 0, i = 1, 2, 3,..., n be real numbers such that Find the maximum value of subject to the constraint Find the maximum value of Let pi > 0, i = 1, 2, 3,..., n be real numbers such that Find the maximum value of subject to the constraint subject to the constraint Let pi > 0, i = 1, 2, 3,..., n be real numbers such that Find the maximum value of subject to the constraint

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