Question

In the following question, find the coordinates of the single stationary point (a, b) of the function:

φ(x, y) = -2x² + xy + 2y² - 4x - 4y.
Partial derivatives for φ(x, y):
∂φ/∂x, ∂φ/∂y.

Answer 1

By taking the partial derivatives of the function and setting them to zero, we solve the system of equations to find that the stationary point of the function φ(x, y) is (0, 1).

Step-by-step explanation:

To determine the stationary point for the function φ(x, y) = -2x² + xy + 2y² - 4x - 4y, we need to find the partial derivatives ∂φ/∂x and ∂φ/∂y, and set them equal to zero. The partial derivatives are:

• ∂φ/∂x = -4x + y - 4
• ∂φ/∂y = x + 4y - 4

Setting these to zero gives us the system of equations:

• -4x + y - 4 = 0
• x + 4y - 4 = 0

Solving this system, we add the first equation to four times the second equation to eliminate y and get:

x = 0

Then we substitute x = 0 into the second equation:

4y - 4 = 0

y = 1

The stationary point of the function is therefore (0, 1).