Find the absolute maximum and minimum values of

f(x, y) = e
−x
2−y
2
(x
2 + 2y
2
) on the disk x
2 + y
2 ≤ 4

To find the absolute maximum and minimum values of the function

\[
f(x, y) = e^{-x^2 - y^2}(x^2 + 2y^2)
\]

on the disk \( x^2 + y^2 \leq 4 \), we need to consider two cases: the interior of the disk and the boundary.

### Step 1: Finding Critical Points in the Interior

First, let’s find the critical points of \( f(x, y) \) by setting the partial derivatives with respect to \( x \) and \( y \) equal to zero.

1. **Partial derivative with respect to \( x \):**
   \[
   f_x = \frac{\partial}{\partial x}\left(e^{-x^2 - y^2}(x^2 + 2y^2)\right)
   \]

   Using the product rule:
   \[
   f_x = e^{-x^2 - y^2}(2x) - e^{-x^2 - y^2}(2x)(x^2 + 2y^2) = 0
   \]

This

Find the absolute maximum and minimum values of f(x, y) = e^(-x^2 - y^2)(x^2 + 2y^2) on the disk x^2 + y^2 ≤ 4

Learn how to find the absolute maximum and minimum values of the function f(x, y) = e^(-x^2 - y^2)(x^2 + 2y^2) on the disk x^2 + y^2 ≤ 4. This problem covers multivariable optimization techniques, critical points, and the application of the Extreme Value Theorem in a bounded region. Suitable for students in college-level calculus courses.

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