Maximize the function subject to the constraint .

Question

Maximize the function  subject to the constraint  .

Quizplus · Accepted Answer

Using Lagrange Multipliers, we need to solve the system of equations:
∇f = λ∇g
g = 0
Where f = 11eaa8b1_d98f_f7b6_b1b6_ffc81cdaea2b_TB6026_11
And g = 11eaa8b1_d990_1ec7_b1b6_d165e9c086fe_TB6026_11
Taking the partial derivative of f with respect to x and y, we get:
fx = 11e^(x+2y)
fy = 22xe^(x+2y)
Taking the partial derivative of g with respect to x and y, we get:
gx = xe^x - y^2
gy = -2xy
Setting the gradients equal to each other and solving for x, y, and λ, we get:
x = 1/4
y = -1/4
λ = e^-5/2
Plugging these values back into f, we get the maximum value to be approximately 3.68, which corresponds to choice C.

Maximize the Function Subject to the Constraint