To classify the critical points of the function $f(x, y) = x^4 + y^4 - 4x^2 - 4y^2$ using the second derivative test, follow these steps:

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Step 1: Find the critical points

1. Compute the partial derivatives of $f(x, y)$: $f_x = \frac{\partial f}{\partial x} = 4x^3 - 8x$ $f_y = \frac{\partial f}{\partial y} = 4y^3 - 8y$

2. Set $f_x = 0$ and $f_y = 0$ to find critical points: $4x(x^2 - 2) = 0 \quad \text{and} \quad 4y(y^2 - 2) = 0$ From each equation, we solve: $x = 0, \quad x = \pm\sqrt{2}, \quad y = 0, \quad y = \pm\sqrt{2}.$

3. The critical points are: $(0, 0), \quad (\sqrt{2}, 0), \quad (-\sqrt{2}, 0), \quad (0, \sqrt{2}), \quad (0, -\sqrt{2}), \quad (\sqrt{2}, \sqrt{2}), \quad (\sqrt{2}, -\sqrt{2}), \quad (-\sqrt{2}, \sqrt{2}), \quad (-\sqrt{2}, -\sqrt{2}).$

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Step 2: Compute the second partial derivatives

1. Compute the second partial derivatives: $f_{xx} = \frac{\partial^2 f}{\partial x^2} = 12x^2 - 8, \quad f_{yy} = \frac{\partial^2 f}{\partial y^2} = 12y^2 - 8, \quad f_{xy} = \frac{\partial^2 f}{\partial x \partial y} = 0.$

2. Construct the Hessian determinant: $H = f_{xx}f_{yy} - (f_{xy})^2 = (12x^2 - 8)(12y^2 - 8) - 0.$

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Step 3: Analyze each critical point

For each critical point, substitute $x$ and $y$ into $f_{xx}$, $f_{yy}$, and $H$ to classify them:

1. At $(0, 0)$: $f_{xx} = -8, \quad f_{yy} = -8, \quad H = (-8)(-8) - 0 = 64 > 0.$ Since $H > 0$ and $f_{xx} < 0$, $(0, 0)$ is a local maximum.

2. At $(\pm\sqrt{2}, 0)$: $f_{xx} = 16, \quad f_{yy} = -8, \quad H = (16)(-8) - 0 = -128 < 0.$ Since $H < 0$, these are saddle points.

3. At $(0, \pm\sqrt{2})$: $f_{xx} = -8, \quad f_{yy} = 16, \quad H = (-8)(16) - 0 = -128 < 0.$ Since $H < 0$, these are saddle points.

4. At $(\pm\sqrt{2}, \pm\sqrt{2})$: $f_{xx} = 16, \quad f_{yy} = 16, \quad H = (16)(16) - 0 = 256 > 0.$ Since $H > 0$ and $f_{xx} > 0$, these are local minima.

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Conclusion

• Local maximum: $(0, 0)$
• Local minima: $(\pm\sqrt{2}, \pm\sqrt{2})$
• Saddle points: $(\pm\sqrt{2}, 0), (0, \pm\sqrt{2})$

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Do you want more details or explanations on any part?

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Related Questions:

1. How does the second derivative test work for functions with mixed partial derivatives?
2. What happens if the Hessian determinant is zero at a critical point?
3. Can you explain how the Hessian matrix is derived for multivariable functions?
4. How do the critical points relate to the graph of $f(x, y)$?
5. What is the physical or geometric interpretation of saddle points?

Tip: Always verify the conditions of the second derivative test carefully, as it may not classify a critical point if $H = 0$.