The function $f(x, y) = e^{x - y} \cdot (x^2 + 2y^2)$ combines an exponential and a polynomial expression. Let’s analyze and explore it step by step:

Properties and Analysis:

1. Domain: The function $f(x, y)$ is defined for all real numbers $x$ and $y$ because both the exponential $e^{x-y}$ and the polynomial $x^2 + 2y^2$ are well-defined for all real $x, y$.

2. Partial Derivatives:

   • To compute $\frac{\partial f}{\partial x}$: $\frac{\partial f}{\partial x} = \frac{\partial}{\partial x} \left( e^{x-y} \cdot (x^2 + 2y^2) \right)$ Using the product rule: $\frac{\partial f}{\partial x} = e^{x-y} \cdot \frac{\partial}{\partial x}(x^2 + 2y^2) + \frac{\partial}{\partial x}(e^{x-y}) \cdot (x^2 + 2y^2)$ Simplifying: $\frac{\partial f}{\partial x} = e^{x-y} \cdot 2x + e^{x-y} \cdot (x^2 + 2y^2)$ $\frac{\partial f}{\partial x} = e^{x-y} \cdot \left(2x + x^2 + 2y^2 \right)$

   • To compute $\frac{\partial f}{\partial y}$: $\frac{\partial f}{\partial y} = \frac{\partial}{\partial y} \left( e^{x-y} \cdot (x^2 + 2y^2) \right)$ Using the product rule: $\frac{\partial f}{\partial y} = e^{x-y} \cdot \frac{\partial}{\partial y}(x^2 + 2y^2) + \frac{\partial}{\partial y}(e^{x-y}) \cdot (x^2 + 2y^2)$ Simplifying: $\frac{\partial f}{\partial y} = e^{x-y} \cdot 4y - e^{x-y} \cdot (x^2 + 2y^2)$ $\frac{\partial f}{\partial y} = e^{x-y} \cdot \left(4y - x^2 - 2y^2\right)$

3. Critical Points: To find critical points, solve $\frac{\partial f}{\partial x} = 0$ and $\frac{\partial f}{\partial y} = 0$ simultaneously.

4. Growth and Decay:

   • $e^{x-y}$ grows or decays exponentially depending on the sign of $x-y$.
   • $x^2 + 2y^2$ grows quadratically, emphasizing the dominance of the $y$-axis due to the coefficient 2.

5. Behavior:

   • $f(x, y)$ will be large in magnitude where $x$ and $y$ are large and of similar values (exponential growth from $e^{x-y}$).
   • Near the origin ($x, y \to 0$), $f(0, 0) = 0$.

Would you like me to analyze further, solve specific problems involving this function, or visualize its graph?

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Related Questions:

1. What is the behavior of $f(x, y)$ as $x, y \to \infty$?
2. How can we classify the critical points of $f(x, y)$?
3. What is the second partial derivative of $f(x, y)$?
4. Can we visualize $f(x, y)$ for specific ranges of $x$ and $y$?
5. How does $f(x, y)$ behave along the lines $x = y$ or $y = 0$?

Tip:

When analyzing multivariable functions, always consider symmetry and boundary behavior to simplify the study of critical points and extrema.