GATE 2024 Mechanical Engineering Q15

Question

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)ᵀp + ½ pᵀ∇²f(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?

Answer

∇f(x₀)ᵀp > 0 and pᵀ∇²f(ψ)p = 0

Answer

∇f(x₀)ᵀp = 0 and pᵀ∇²f(ψ)p > 0

Answer

∇f(x₀)ᵀp = 0 and pᵀ∇²f(ψ)p = 0

Answer

∇f(x₀)ᵀp = 0 and pᵀ∇²f(ψ)p < 0