Use Lagrange multipliers to find maximum and minimum values of subject to .
A) The maximum value is 0, and there is no minimum value. 2
B) The maximum value is , and the minimum value is
C) The maximum value is , and the minimum value is
D) The maximum value is , and the minimum value is
E) There is no maximum value, and the minimum value is 0.

Question

Use Lagrange multipliers to find maximum and minimum values of  subject to  .
A) The maximum value is 0, and there is no minimum value. 2 
B) The maximum value is , and the minimum value is 
C) The maximum value is , and the minimum value is 
D) The maximum value is , and the minimum value is 
E) There is no maximum value, and the minimum value is 0.

Quizplus · Accepted Answer

We need to use Lagrange multipliers to find the maximum and minimum values of 11ead0bc_8daf_f79f_99a0_1b402a1c147e_TB6988_11 subject to the constraint 11ead0bc_8daf_f7a0_99a0_7fb7d7245aea_TB6988_11. Let x be the value of 11ead0bc_8daf_f79f_99a0_1b402a1c147e_TB6988_11 and y be the value of 11ead0bc_8daf_f7a0_99a0_7fb7d7245aea_TB6988_11. Then we want to maximize and minimize x subject to the constraint y=11ead0bc_8daf_f7a0_99a0_7fb7d7245aea_TB6988_11. We set up the following equation using Lagrange multipliers: ∇f(x,y) = λ∇g(x,y) where f(x,y) = 11ead0bc_8daf_f79f_99a0_1b402a1c147e_TB6988_11 and g(x,y) = y - 11ead0bc_8daf_f7a0_99a0_7fb7d7245aea_TB6988_11. So, ∇f(x,y) = <1,0> and ∇g(x,y) = <0,1>, and we get the equation: <1,0> = λ<0,1> This gives us λ = 0, which means there are critical points at all points where y=11ead0bc_8daf_f7a0_99a0_7fb7d7245aea_TB6988_11. To determine if these are maximum or minimum points, we look at the second derivative test. Taking the second partial derivative of f: ∂2f/∂x2 = 0, ∂2f/∂y2 = 0, ∂2f/∂x∂y = 0 This gives us the Hessian matrix: H = [[0, 0], [0, 0]] Since the Hessian matrix is indefinite, the function has no maximum or minimum values. Therefore, the maximum value is not defined, and the minimum value occurs at one of the critical points which is at y=11ead0bc_8daf_f7a0_99a0_7fb7d7245aea_TB6988_11. The minimum value occurs when x is the minimum value of the function over its domain, which is 11ead0bc_8db0_45c2_99a0_b9f3118ac3c2_TB6988_11. So the answer is B: the maximum value is not defined, and the minimum value is 11ead0bc_8db0_45c2_99a0_b9f3118ac3c2_TB6988_11.

Use Lagrange Multipliers to Find Maximum and Minimum Values of