Use Lagrange multipliers to find the maximum and minimum values of subject to .
A) The maximum value is 0, and there is no minimum value.
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 the maximum and minimum values of  subject to  .
A) The maximum value is 0, and there is no minimum value. 
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 solve this problem. Let

f(x,y) = y
g(x,y) = x^2 + y^2 - 1.

Then, solving the system of equations

∇f = λ∇g 
g(x,y) = 0

gives us

2λx = 0 
2λy = 1 
x^2 + y^2 - 1 = 0.

From the first equation, we know that x = 0 or λ = 0. If λ = 0, then y = 0 and x^2 = 1, so x = ±1. If x = 0, then y = 1 or -1. Therefore, the critical points are (0,1), (0,-1), (-1/√2,1/√2), and (1/√2,-1/√2).

Now, we need to check the value of f at each of these points to determine the maximum and minimum values.

f(0,1) = 1 
f(0,-1) = -1 
f(-1/√2,1/√2) = 1/√2 
f(1/√2,-1/√2) = -1/√2

Therefore, the maximum value of f is 1/√2, and the minimum value of f is -1. So the answer is C.

Use Lagrange Multipliers to Find the Maximum and Minimum Values