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The HESSIAN of a twice-differentiable scalar function f: ℝⁿ → ℝ is
AThe trace of the Jacobian
BThe n×n matrix of second partial derivatives ∂²f/∂x_i ∂x_j
CThe n-vector of first partials (this is the gradient)
DA scalar second derivative
Answer & Solution
Correct answer: B. The n×n matrix of second partial derivatives ∂²f/∂x_i ∂x_j
Hessian H = [∂²f/∂x_i ∂x_j]_(i,j) — symmetric for sufficiently smooth f (Schwarz's theorem). Positive-definite Hessian ⇒ local minimum at critical points.
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