The exercise involves determining critical points and their nature for various functions using principles of multivariable calculus and optimization. Here's a structured explanation for the first two functions. Further solutions can be computed similarly:

1. $f(x, y) = x^4 + y^4 - 2(x - y)^2$

• Gradient computation: $\frac{\partial f}{\partial x} = 4x^3 - 4(x - y), \quad \frac{\partial f}{\partial y} = 4y^3 + 4(x - y)$

• Set gradient equal to zero: $4x^3 - 4(x - y) = 0 \quad \text{and} \quad 4y^3 + 4(x - y) = 0$ Simplify: $x^3 = x - y, \quad y^3 = -(x - y)$ Solving these equations yields the critical points, such as $(0, 0)$ and symmetric solutions.

• Hessian matrix: Compute the second partial derivatives: $f_{xx} = 12x^2 - 4, \quad f_{yy} = 12y^2 - 4, \quad f_{xy} = -4$ Hessian: $H = \begin{bmatrix} 12x^2 - 4 & -4 \\ -4 & 12y^2 - 4 \end{bmatrix}$ Evaluate eigenvalues of $H$ at each critical point to classify as minimum, maximum, or saddle point.

---

2. $f(x, y, z) = x^2 + xyz - z + y$

• Gradient computation: $\frac{\partial f}{\partial x} = 2x + yz, \quad \frac{\partial f}{\partial y} = xz + 1, \quad \frac{\partial f}{\partial z} = xy - 1$

• Set gradient equal to zero: $2x + yz = 0, \quad xz + 1 = 0, \quad xy - 1 = 0$ Solve these equations simultaneously to find critical points.

• Hessian matrix: Compute the second partial derivatives: $f_{xx} = 2, \quad f_{yy} = 0, \quad f_{zz} = 0, \quad f_{xy} = z, \quad f_{xz} = y, \quad f_{yz} = x$ Hessian: $H = \begin{bmatrix} 2 & z & y \\ z & 0 & x \\ y & x & 0 \end{bmatrix}$ Use eigenvalues of $H$ at critical points for classification.

---

Would you like me to complete the analysis for all functions or provide detailed steps for specific ones? Let me know how you'd like to proceed!

Relative Questions:

1. How do the eigenvalues of the Hessian help in classifying critical points?
2. What are sufficient conditions for a point to be a local maximum in multivariable optimization?
3. How is the method of Lagrange multipliers applied in problems with constraints?
4. Can critical points exist outside the domain of a given function? Why or why not?
5. How do convexity properties of functions affect the classification of critical points?

Tip:

Always verify if the Hessian determinant is zero before using eigenvalues to classify the critical points, as this could indicate a degenerate case.