Questions: Find the maximum and minimum values of the function f(x, y)=x^2 y subject to 3 x^2+2 y^2=18 Please show your answers to at least 4 decimal places. Enter DNE if the value does not exist.

Find the maximum and minimum values of the function f(x, y)=x^2 y subject to 3 x^2+2 y^2=18
Please show your answers to at least 4 decimal places. Enter DNE if the value does not exist.

Transcript text: Find the maximum and minimum values of the function $f(x, y)=x^{2} y$ subject to $3 x^{2}+2 y^{2}=18$ Please show your answers to at least 4 decimal places. Enter DNE if the value does not exist. Maximum value: Minimum value:

Solution

Solution Steps

Step 1: Formulate the Lagrange function

L(x, y, lambda) = x^2_y - lambda(3_x^2 + 2*y^2 - 18)

Step 2: Compute the partial derivatives and set them to zero

∂L/∂x = 0: - 6 \lambda x + 2 x y = 0 ∂L/∂y = 0: - 4 \lambda y + x^{2} = 0 ∂L/∂lambda = 0: - 3 x^{2} - 2 y^{2} + 18 = 0

Step 3: Solve the system of equations

Solution 1: x = 0, y = -3, lambda = 0 Solution 2: x = 0, y = 3, lambda = 0 Solution 3: x = -2, y = -1.732, lambda = -0.577 Solution 4: x = 2, y = -1.732, lambda = -0.577 Solution 5: x = -2, y = 1.732, lambda = 0.577 Solution 6: x = 2, y = 1.732, lambda = 0.577

Step 4: Evaluate the function at the solutions to find the extrema

Value 1: f(x, y) = 0 Value 2: f(x, y) = 0 Value 3: f(x, y) = -6.928 Value 4: f(x, y) = -6.928 Value 5: f(x, y) = 6.928 Value 6: f(x, y) = 6.928