Optimization of f(x, y) = e^(-x^2 - y^2)(x^2 + 2y^2) on the Disk x^2 + y^2 ≤ 4

Math Problem Statement

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.

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.

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: $f_x = e^{-x^2 - y^2}(2x) - e^{-x^2 - y^2}(2x)(x^2 + 2y^2) = 0$

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Multivariable Calculus
Optimization
Critical Points
Partial Derivatives

f(x, y) = e^(-x^2 - y^2)(x^2 + 2y^2)
Critical point condition: ∂f/∂x = 0, ∂f/∂y = 0

Extreme Value Theorem
Optimization on a Closed Region

College/University Level (Calculus II/III)

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Local Maxima, Minima, and Saddle Points for f(x, y) = e^{-(x^2 + y^2 - 4y)}

Finding Absolute Maximum and Minimum Using Lagrange Multipliers

Find Maximum and Minimum of f(x, y) = e^(-xy/16) with Constraint