Question

Determine the nature of the stationary points to the function z=2xy(xy+2y)-4y(y^(2)-4) .

Answer 1

To determine the nature of the stationary points of the function, we can follow these steps: find the partial derivatives, solve the system of equations, and calculate the second partial derivatives.

Step-by-step explanation:

To determine the nature of the stationary points of the function, we need to find the critical points by taking the partial derivatives of z with respect to x and y, and setting them equal to zero. Then, we can use the second partial derivative test to determine the nature of the critical points. Let's go through the steps:

1. Find the partial derivative of z with respect to x: ∂z/∂x = 2y(xy + 2y) - 4y(y^2 - 4) = 2y^2 + 4y^2 - 4y^3 + 16y
2. Find the partial derivative of z with respect to y: ∂z/∂y = 2x(xy + 2y) - 4(y^2 - 4) = 2xy^2 + 4xy + 8y - 4y^2 + 16
3. Solve the system of equations ∂z/∂x = 0 and ∂z/∂y = 0 to find the critical points.
4. Calculate the second partial derivatives to determine the nature of the critical points.