Question

Let f(x, y) = (x² + y²) * e^(-(x-y)).
Find all critical points of f(x, y).

Answer 1

To find the critical points of the function f(x, y) = (x² + y²) * e^(-(x-y)), you need to take the partial derivatives of f with respect to x and y, set them equal to zero, and solve the resulting equations.

Step-by-step explanation:

The critical points of the function f(x, y) = (x² + y²) * e^(-(x-y)) can be found by taking the partial derivatives of f with respect to x and y and setting them equal to zero. Let's find the critical points step-by-step:

1. Take the partial derivative of f(x, y) with respect to x: ∂f/∂x = 2x * e^(-(x-y)) - (x² + y²) * e^(-(x-y))
2. Take the partial derivative of f(x, y) with respect to y: ∂f/∂y = -2y * e^(-(x-y)) - (x² + y²) * e^(-(x-y))
3. Set both derivatives equal to zero and solve the resulting equations:

• 2x * e^(-(x-y)) - (x² + y²) * e^(-(x-y)) = 0
• -2y * e^(-(x-y)) - (x² + y²) * e^(-(x-y)) = 0

Solve the system of equations to find the critical points.