Math Problem Statement

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Solution

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Math Problem Analysis

Mathematical Concepts

Critical points
Hessian matrix
Gradient vector
Optimization
Multivariable calculus

Formulas

f(x) ≈ f(x₀) + 1/2 f''(x₀) (x - x₀)²
Hessian matrix: Hf = [[fxx, fxy], [fxy, fyy]]
Gradient: ∇f(x₀, y₀) = [fx(x₀, y₀), fy(x₀, y₀)]

Theorems

Second derivative test for critical points
Gradient flow method for local maxima

Suitable Grade Level

Undergraduate (Calculus III or Multivariable Calculus)