find and classify all local minima, local maxima, and saddle points of the function f(x,y)= e^-y(x^2+y^2)
To find and classify the local minima, local maxima, and saddle points of the function f(x, y) = e^(-y(x^2 + y^2)), you need to perform the following steps:
Step 1: Find the partial derivatives of f(x, y) with respect to x and y.
To find the partial derivative with respect to x, you treat y as a constant and differentiate only the term involving x, while keeping the e^(-y(x^2 + y^2)) intact. The partial derivative of f(x, y) with respect to x is given by:
∂f/∂x = -2xye^(-y(x^2 + y^2))
To find the partial derivative with respect to y, you treat x as a constant and differentiate only the term involving y, while keeping the e^(-y(x^2 + y^2)) intact. The partial derivative of f(x, y) with respect to y is given by:
∂f/∂y = -e^(-y(x^2 + y^2))(1 + 2y^2 - 2x^2y)
Step 2: Set the partial derivatives equal to zero and solve for x and y to find the critical points.
To find the critical points, we set both partial derivatives equal to zero and solve the resulting system of equations:
-2xye^(-y(x^2 + y^2)) = 0
-e^(-y(x^2 + y^2))(1 + 2y^2 - 2x^2y) = 0
Solving these equations simultaneously may seem difficult analytically, so let's use numerical methods or graphing tools to find the critical points.
Step 3: Classify the critical points using the second partial derivative test.
Once you have identified the critical points, you can classify them using the second partial derivative test. To do this, we need to find the second partial derivatives of f(x, y) with respect to x and y.
The second partial derivative with respect to x, denoted as ∂²f/∂x² (read as "the second partial derivative of f with respect to x squared"), is given by:
∂²f/∂x² = -2ye^(-y(x^2 + y^2)) + 4x^2y^2e^(-y(x^2 + y^2))
The second partial derivative with respect to y, denoted as ∂²f/∂y² (read as "the second partial derivative of f with respect to y squared"), is given by:
∂²f/∂y² = 2e^(-y(x^2 + y^2))(1 + 4y^2 - 4x^2y) - 2x^2e^(-y(x^2 + y^2))
The mixed partial derivative with respect to x and y, denoted as ∂²f/∂x∂y (read as "the second partial derivative of f with respect to x and then y" or "the partial derivative of (∂f/∂x) with respect to y"), is given by:
∂²f/∂x∂y = 2xye^(-y(x^2 + y^2))(2y - 2x^2)
Step 4: Evaluate the second partial derivatives at each critical point and classify them.
At each critical point (x, y), evaluate the second partial derivatives ∂²f/∂x², ∂²f/∂y², and ∂²f/∂x∂y:
- If ∂²f/∂x² > 0 and ∂²f/∂y² > 0, then it is a local minimum.
- If ∂²f/∂x² < 0 and ∂²f/∂y² < 0, then it is a local maximum.
- If ∂²f/∂x² and ∂²f/∂y² have opposite signs (one positive and the other negative) or if ∂²f/∂x∂y ≠ 0, then it is a saddle point.
Note: If any of the second partial derivatives are equal to zero, the test is inconclusive, and further investigation may be required.
By following these four steps, you can find and classify the local minima, local maxima, and saddle points of the function f(x, y) = e^(-y(x^2 + y^2)).