Identify the critical point of the function f(x,y)=x^2-4x-2y+y^2

To identify the critical points of the function \( f(x, y) = x^2 - 4x - 2y + y^2 \), we follow these steps:

### 1. Compute the partial derivatives.
The critical points occur where the first partial derivatives of \( f(x, y) \) are equal to zero.

\[
f_x = \frac{\partial f}{\partial x} = 2x - 4, \quad f_y = \frac{\partial f}{\partial y} = 2y - 2
\]

### 2. Set the partial derivatives to zero.
\[
f_x = 0 \implies 2x - 4 = 0 \implies x = 2
\]
\[
f_y = 0 \implies 2y - 2 = 0 \implies y = 1
\]

### 3. Solve the system of equations.
The solution to \( x = 2 \) and \( y = 1 \) gives the critical point:
\[
(x, y) = (2, 1)
\]

### 4. Verify the result.
Substitute \( x = 2 \) and \( y = 1 \) back into \( f(x, y) \):
\[
f(2, 1) = (2)^2 - 4(2) - 2(1) + (1)^2 = 4 - 8 - 2 + 1 = -5
\]
Thus, the function has a critical point at \( (2, 1) \) with \( f(2, 1) = -5 \).

Would you like to analyze the nature of this critical point (e.g., minimum, maximum, saddle point) using the second derivative test? 

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### 5 Related Questions:
1. What is the second derivative test, and how is it applied in two-variable functions?
2. Can you find the critical points of \( f(x, y) = x^2 + y^2 - 6x + 4y + 9 \)?
3. How does the behavior of a quadratic function change with its critical points?
4. What are some applications of critical points in optimization problems?
5. What is the significance of the determinant of the Hessian matrix in analyzing critical points?

### Tip:
Always confirm the nature of a critical point using the second derivative test or equivalent criteria to ensure its classification (e.g., local minimum, maximum, or saddle point).

Learn how to find the critical point of the function f(x, y) = x^2 - 4x - 2y + y^2 using partial derivatives. This detailed solution covers finding critical points, verifying results, and includes tips for analyzing the nature of critical points. Ideal for undergraduate calculus students.

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