Mathematics · Advanced Calculus and Optimization
Let f(.) be a twice differentiable function from ℝ2 → ℝ. For small ||p||, Taylor's expansion gives f(x0 + p) = f(x0) + ∇f(x0)Tp + 12pT∇2f(ψ)p, where ψ is on the line segment joining x0 and x0 + p. If x0 is a strict local minimum of f(x), which statement is TRUE?
Verified Solution4 OptionsUpdated: October 2026👁️1 View
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- 1∇f(x0)Tp = 0 and pT∇2f(ψ)p < 0
- 2∇f(x0)Tp > 0 and pT∇2f(ψ)p = 0
- 3∇f(x0)Tp = 0 and pT∇2f(ψ)p = 0
- 4∇f(x0)Tp = 0 and pT∇2f(ψ)p > 0✓
Correct answer
∇f(x0)Tp = 0 and pT∇2f(ψ)p > 0
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