Question

Find the local extrema of the function f(x,y) = x² y² + 2xy?

Answer 1

To find the local extrema of the given function, take partial derivatives with respect to x and y, set them equal to zero, solve the resulting system of equations, and use the second partial derivative test. The critical points are the local extrema.

Step-by-step explanation:

The function f(x, y) = x²y² + 2xy represents a surface in three-dimensional space. To find the local extrema of this function, we need to find the critical points by taking the partial derivatives with respect to x and y, setting them equal to zero, and solving the resulting system of equations.

1. Take the partial derivative of f(x, y) with respect to x: ∂f/∂x = 2xy² + 2y.
2. Take the partial derivative of f(x, y) with respect to y: ∂f/∂y = 2x²y + 2x.
3. Set both partial derivatives equal to zero and solve the system of equations to find the critical points.
4. To determine if a critical point is a local minimum, local maximum, or saddle point, we can use the second partial derivative test.
5. Take the second partial derivatives of f(x, y) with respect to x and y: ∂²f/∂x² = 2y² and ∂²f/∂y² = 2x².
6. Evaluate the second partial derivatives at each critical point.
7. If the second partial derivatives are both positive, the critical point is a local minimum. If they are both negative, the critical point is a local maximum. If they have opposite signs, the critical point is a saddle point.