GATE ME 2024 Set 1 — Question 15
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Engineering Mathematics → Multivariable Calculus → Maxima & Minima (Several Variables)
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Question
Let be a twice differentiable function from . If where is sufficiently small (here is the Euclidean norm or distance function), then where oldsymbol{\psi} \in \mathbb{R}^2 is a point on the line segment joining and . If is a strict local minimum of , then which one of the following statements is TRUE?
Correct answer
(B) f(x₀)^T p = 0 and p^T ² f() p 0
Solution
For a function to have a local minimum at , the first-order necessary condition is that the gradient at that point must be zero:
for any vector .For to be a strict local minimum, we must have for all sufficiently small non-zero vectors .
Using the given Taylor expansion:
Since , the condition becomes:
.Therefore, the correct statement is and .
for any vector .For to be a strict local minimum, we must have for all sufficiently small non-zero vectors .
Using the given Taylor expansion:
Since , the condition becomes:
.Therefore, the correct statement is and .
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