Classify all critical points of the function f(x,y)=x^2-y^2+4x-2y+1.

I know I have to take the partial derivative and set them equal to zero to get my points and then use the Hessian. For some reason I am not getting the correct answer.
partial d/dx 2x+4 and partial d/dy 2y-2 and solving each for zero gives me x=-2 and y=1

To classify the critical points of the function f(x, y) = x^2 - y^2 + 4x - 2y + 1, you need to find the critical points by equating the partial derivatives to zero and then use the Hessian matrix to determine their nature.

Taking the partial derivative with respect to x, you correctly found:

∂f/∂x = 2x + 4

Setting this equal to zero, we have:
2x + 4 = 0
2x = -4
x = -2

Next, taking the partial derivative with respect to y, you correctly found:

∂f/∂y = 2y - 2

Setting this equal to zero, we have:
2y - 2 = 0
2y = 2
y = 1

So far, your critical points are (x, y) = (-2, 1).

Now, let's calculate the Hessian matrix:

H = | ∂^2f/∂x^2 ∂^2f/(∂x∂y) |
| ∂^2f/(∂y∂x) ∂^2f/∂y^2 |

To do this, we need to find the second partial derivatives:

∂^2f/∂x^2 = 2
∂^2f/(∂x∂y) = 0
∂^2f/(∂y∂x) = 0
∂^2f/∂y^2 = -2

Now, let's substitute these values into the Hessian matrix:

H = | 2 0 |
| 0 -2 |

To classify the critical points using the Hessian, we need to determine the eigenvalues of the Hessian matrix.

The determinant of H is:
det(H) = 2(-2) - 0(0) = -4

Since the determinant is negative, we know that the critical point is a saddle point.

Therefore, the critical point of the function f(x, y) = x^2 - y^2 + 4x - 2y + 1 is a saddle point at (x, y) = (-2, 1).

To classify the critical points of a function, you need to find the values of x and y where the partial derivatives are equal to zero, as you correctly mentioned. Let's go through the process step by step.

1. Find the partial derivative with respect to x, denoted as ∂f/∂x:
∂f/∂x = 2x + 4

2. Set the partial derivative equal to zero and solve for x:
2x + 4 = 0
2x = -4
x = -2

So, one critical point is x = -2.

3. Find the partial derivative with respect to y, denoted as ∂f/∂y:
∂f/∂y = -2y - 2

4. Set the partial derivative equal to zero and solve for y:
-2y - 2 = 0
-2y = 2
y = -1

Another critical point is y = -1.

Now, we have two critical points: (-2, -1). To determine the nature of these critical points (maximum, minimum, or saddle point), you need to use the Hessian matrix.

1. Compute the second partial derivatives:
∂²f/∂x² = 2
∂²f/∂y² = -2

2. Compute the mixed partial derivatives:
∂²f/∂x∂y = 0
∂²f/∂y∂x = 0

3. Construct the Hessian matrix:
H = [∂²f/∂x² ∂²f/∂x∂y]
[∂²f/∂y∂x ∂²f/∂y² ]

The Hessian matrix for this function is:
H = [2, 0]
[0, -2]

4. Determine the eigenvalues of the Hessian matrix. The eigenvalues will help determine the nature of the critical points.

The eigenvalues of the Hessian matrix are:
λ1 = 2
λ2 = -2

Since both eigenvalues have different signs, we can conclude that the critical point (-2, -1) is a saddle point.

Therefore, the critical point of the function f(x, y) = x^2 - y^2 + 4x - 2y + 1 is a saddle point at (-2, -1).