Find the maximum and minimum values of the function
Subject to the constraint .

Question

Find the maximum and minimum values of the function ​  ​
Subject to the constraint  .
​

Quizplus · Accepted Answer

To find the maximum and minimum values of the function subject to the constraint, we need to use the method of Lagrange multipliers. Let f(x,y) = 11x^2 + 11y^2 and g(x,y) = x^2 + y^2 - 1.

We set up the Lagrange function L(x,y,λ) = f(x,y) - λg(x,y) and solve the system of equations:

∂L/∂x = 22x - 2λx = 0
∂L/∂y = 22y - 2λy = 0
∂L/∂λ = x^2 + y^2 - 1 = 0

From the first two equations, we get 22x = 2λx and 22y = 2λy, which gives λ = 11. Substituting this into the third equation, we get x^2 + y^2 = 1, which is the constraint.

Therefore, the critical points are (±1/√22, ±1/√22). To determine which of these points correspond to maximum or minimum values of f, we need to evaluate f at each of these points.
f(1/√22,1/√22) = f(-1/√22,-1/√22) = 11/22
f(1/√22,-1/√22) = f(-1/√22,1/√22) = -11/22

Therefore, the maximum value of f subject to the constraint is 11/22 and the minimum value is -11/22. The answer is therefore C, with the correct maxima and minima values.

Find the Maximum and Minimum Values of the Function ​

Find the Maximum and Minimum Values of the Function