Question

The stationary points of the function f(x, y) = xy^2 + x^2 + 2y^2 - 8 are: (a) (0, 0), (-2,...

Answer 1

To identify the stationary points of the function f(x, y) = xy^2 + x^2 + 2y^2 - 8, calculate the partial derivatives with respect to x and y, set them to zero, and solve the resulting system of equations.

Step-by-step explanation:

To find the stationary points of a bivariate function like f(x, y) = xy^2 + x^2 + 2y^2 - 8, we need to take the partial derivatives of the function with respect to x and y and set them equal to zero.

Finding the Partial Derivatives:

1. For ∂f/∂x (partial derivative of f with respect to x):

• f_x = y^2 + 2x

2. For ∂f/∂y (partial derivative of f with respect to y):

• f_y = 2xy + 4y

Setting the Partial Derivatives Equal to Zero:

To find the stationary points, we need to solve the system of equations where f_x = 0 and f_y = 0.

Solving the System:

1. Set y^2 + 2x = 0
2. Set 2xy + 4y = 0

From these equations, we can determine the points where the function does not increase or decrease, thus identifying the stationary points.