The function

f(x,y)=(x2+y2)2−8(x2+y2)+8xy
 

has stationary points at some of the following points, (x,y)
. In each case identify whether the point is stationary, and if so find out if it is a maximum, minimum or saddle point.

The point (0,0)
 is Answer 1 Question 2
 
The point (1,1)
 is Answer 2 Question 2
 
The point (−1,−1)
 is Answer 3 Question 2
 
The point (−3–√,3–√)
 is Answer 4 Question 2
 
The point (3–√,−3–√)
 is

To determine whether each of the given points is a stationary point for the function \( f(x,y) = (x^2 + y^2)^2 - 8(x^2 + y^2) + 8xy \), we need to:

1. Compute the first partial derivatives \( f_x \) and \( f_y \) and set them equal to zero to find the stationary points.
2. Evaluate the second partial derivatives \( f_{xx} \), \( f_{yy} \), and \( f_{xy} \) at each stationary point.
3. Use the second derivative test to classify the stationary points as a maximum, minimum, or saddle point.

### Step 1: Compute the First Partial Derivatives
Let's compute the first partial derivatives of \( f(x, y) \):

\[
f_x = \frac{\partial f}{\partial x}
\]
\[
f_y = \frac{\partial f}{\partial y}
\]

Given \( f(x,y) = (x^2 + y^2)^2 - 8(x^2 + y^2) + 8xy \):

\[
f_x = \frac{\partial}{\partial x}\left[(x^2 + y^2)^2 - 8(x^2 + y^2) + 8xy\right]
\]

\[
f_y = \frac{\partial}{\partial y}\left[(x^2 + y^2)^2 - 8(x^2 + y^2) + 8xy\right]
\]

First, simplify and differentiate with respect to \( x \) and \( y \).

### Step 2: Set the Derivatives Equal to Zero
After finding \( f_x \) and \( f_y \), set each derivative equal to zero to find potential stationary points.

### Step 3: Second Derivative Test
For each stationary point \((x_0, y_0)\), calculate:

\[
f_{xx} = \frac{\partial^2 f}{\partial x^2}, \quad f_{yy} = \frac{\partial^2 f}{\partial y^2}, \quad \text{and} \quad f_{xy} = \frac{\partial^2 f}{\partial x \partial y}
\]

Then, compute the determinant of the Hessian matrix \( H \) at \((x_0, y_0)\):

\[
D = f_{xx}f_{yy} - (f_{xy})^2
\]

- If \( D > 0 \) and \( f_{xx} > 0 \), \((x_0, y_0)\) is a local minimum.
- If \( D > 0 \) and \( f_{xx} < 0 \), \((x_0, y_0)\) is a local maximum.
- If \( D < 0 \), \((x_0, y_0)\) is a saddle point.
- If \( D = 0 \), the test is inconclusive.

Let’s go through the calculations. I'll start by computing the first partial derivatives.Here are the results for the points provided:

1. **Point (0, 0)**:
   - This is a **stationary point** and is classified as a **local maximum**.

2. **Point (1, 1)**:
   - This is a **stationary point** and is classified as a **saddle point**.

3. **Point (-1, -1)**:
   - This is a **stationary point** and is classified as a **saddle point**.

4. **Point (-√3, √3)**:
   - This is a **stationary point** and is classified as a **local minimum**.

5. **Point (√3, -√3)**:
   - This is a **stationary point** and is classified as a **local minimum**.

Would you like more details or have any questions about these results?

Here are 8 related questions to expand on this topic:
1. How is the Hessian matrix used in the second derivative test?
2. What are the general conditions to identify a saddle point?
3. How do stationary points relate to the concept of gradients?
4. Can a point be a local minimum in one direction and a local maximum in another?
5. What does it mean when the Hessian determinant is zero?
6. How can we visualize the function \( f(x, y) \) and its stationary points?
7. What role do the second partial derivatives play in determining concavity?
8. How would these results change if we altered the coefficients in the function?

**Tip:** When analyzing functions of multiple variables, always compute both first and second derivatives to fully understand the behavior of the function at different points.

Learn how to identify stationary points of the multivariable function f(x, y) = (x^2 + y^2)^2 - 8(x^2 + y^2) + 8xy. Explore detailed solutions for points (0,0), (1,1), (-1,-1), (-√3, √3), and (√3, -√3), including classifications as local maximum, local minimum, or saddle point.

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