Find and classify all local minima, local maxima, and saddle points of the function f(x,y)= -3yx^2-3xy^2+36xy
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To find and classify the local minima, local maxima, and saddle points of the function f(x, y) = -3yx^2 - 3xy^2 + 36xy, we need to follow these steps:
Step 1: Find the partial derivatives with respect to x and y.
Step 2: Solve the system of equations obtained by setting both partial derivatives equal to zero.
Step 3: Identify the critical points found in step 2.
Step 4: Classify each critical point as a local minimum, local maximum, or saddle point using the Hessian matrix.
Let's go through each step in detail:
Step 1: Find the partial derivatives with respect to x and y.
The partial derivative of f(x, y) with respect to x, denoted as ∂f/∂x, is obtained by differentiating each term of the function with respect to x while treating y as a constant. Similarly, the partial derivative with respect to y, denoted as ∂f/∂y, is obtained by differentiating each term of the function with respect to y while treating x as a constant.
∂f/∂x = -6xy - 3y^2 + 36y
∂f/∂y = -6yx - 6xy + 36x
Step 2: Solve the system of equations obtained by setting both partial derivatives equal to zero.
Setting ∂f/∂x = 0 and ∂f/∂y = 0, we obtain the following system of equations:
-6xy - 3y^2 + 36y = 0 -- (Equation 1)
-6yx - 6xy + 36x = 0 -- (Equation 2)
Step 3: Identify the critical points found in step 2.
Solving the system of equations (Equation 1 and Equation 2) will give us the critical points.
We can rearrange Equation 1 to solve for x in terms of y:
-6xy + 36y - 3y^2 = 0
-6x(y-6) - 3y^2 = 0
x = (36y - 3y^2) / (6(y-6))
Similarly, rearranging Equation 2, we can solve for y in terms of x:
y = (36x) / (6x + 6x)
Step 4: Classify each critical point as a local minimum, local maximum, or saddle point using the Hessian matrix.
To classify each critical point, we need to use the Hessian matrix. The Hessian matrix of a function is obtained by taking the second partial derivatives of the function with respect to x and y.
The Hessian matrix for our function f(x, y) is as follows:
H = | ∂²f/∂x² ∂²f/∂x∂y |
| ∂²f/∂y∂x ∂²f/∂y² |
Compute the second partial derivatives of f(x, y):
∂²f/∂x² = -6y
∂²f/∂x∂y = -6x - 6y
∂²f/∂y∂x = -6x - 6y
∂²f/∂y² = -6x
Substitute the critical points from step 3 into the Hessian matrix:
For each critical point (x0, y0), substitute x0 for x and y0 for y in the Hessian matrix.
For example, let's substitute the values of x and y when x = (36y - 3y^2) / (6(y-6)) and y = (36x) / (6x + 6y) into the Hessian matrix.
H1 = | -6y0 -6x0 - 6y0 |
| -6x0 - 6y0 -6x0 |
Similarly, substitute the critical point values obtained from other equations into the Hessian matrix to compute Hessian matrices for other critical points.
To classify each critical point, we use the eigenvalues of each Hessian matrix. Depending on the eigenvalues, we can classify the critical point as follows:
1. Local Minimum: If all eigenvalues are positive.
2. Local Maximum: If all eigenvalues are negative.
3. Saddle Point: If there are both positive and negative eigenvalues.
Find the eigenvalues for each Hessian matrix and classify the critical points accordingly.
Note: If the Hessian matrix is not invertible, i.e., it has zero eigenvalues, the method is inconclusive, and further tests may be required.
By following these steps, you should be able to find and classify the local minima, local maxima, and saddle points of the given function.