Mechanical Engineering > Gate 2024 > Differential Equation
Let f(⋅) be a twice differentiable function from ℝ² → ℝ. If p, x0 ∈ ℝ² where ‖ p ‖ is sufficiently small (here ‖ ⋅ ‖ is the Euclidean norm or distance function), then:

f(x0 + p) = f(x0) + ∇f(x0)T p + (1/2) pT ∇² f(ψ) p

where ψ ∈ ℝ² is a point on the line segment joining x0 and x0 + p. If x0 is a strict local minimum of f(x), then which one of the following statements is TRUE?

A
∇f(x0)T p > 0 and pT ∇² f(ψ) p = 0
B
∇f(x0)T p = 0 and pT ∇² f(ψ) p > 0
C
∇f(x0)T p = 0 and pT ∇² f(ψ) p = 0
D
∇f(x0)T p = 0 and pT ∇² f(ψ) p < 0

Correct : a

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