The GATE Grind

GATE 2024 ME – Question 15

Engineering Mathematics · Calculus: Partial derivatives, Taylor series, maxima and minima, Fourier series · 1 mark · Multiple choice

Let $f(\cdot)$ be a twice differentiable function from $\mathbb{R}^2 \to \mathbb{R}$. If $\boldsymbol{p}, \boldsymbol{x_0} \in \mathbb{R}^2$ where $\|\boldsymbol{p}\|$ is sufficiently small (here $\|\cdot\|$ is the Euclidean norm or distance function), then $f(\boldsymbol{x_0} + \boldsymbol{p}) = f(\boldsymbol{x_0}) + \nabla f(\boldsymbol{x_0})^T\boldsymbol{p} + \frac{1}{2}\boldsymbol{p}^T\nabla^2 f(\boldsymbol{\psi})\boldsymbol{p}$ where $\boldsymbol{\psi} \in \mathbb{R}^2$ is a point on the line segment joining $\boldsymbol{x_0}$ and $\boldsymbol{x_0} + \boldsymbol{p}$. If $\boldsymbol{x_0}$ is a strict local minimum of $f(\boldsymbol{x})$, then which one of the following statements is TRUE?

  1. $\nabla f(\mathbf{x_0})^T\boldsymbol{p} > 0$ and $\boldsymbol{p}^T\nabla^2 f(\boldsymbol{\psi})\boldsymbol{p} = 0$
  2. $\nabla f(\mathbf{x_0})^T\boldsymbol{p} = 0$ and $\boldsymbol{p}^T\nabla^2 f(\boldsymbol{\psi})\boldsymbol{p} > 0$
  3. $\nabla f(\mathbf{x_0})^T\boldsymbol{p} = 0$ and $\boldsymbol{p}^T\nabla^2 f(\boldsymbol{\psi})\boldsymbol{p} = 0$
  4. $\nabla f(\mathbf{x_0})^T\boldsymbol{p} = 0$ and $\boldsymbol{p}^T\nabla^2 f(\boldsymbol{\psi})\boldsymbol{p} < 0$

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Correct answer: (B) $\nabla f(\mathbf{x_0})^T\boldsymbol{p} = 0$ and $\boldsymbol{p}^T\nabla^2 f(\boldsymbol{\psi})\boldsymbol{p} > 0$

Explanation

At a local minimum the gradient is zero, so $\nabla f(\mathbf{x_0})^T\boldsymbol{p} = 0$ for every $\boldsymbol{p}$. The expansion then reduces to $f(\mathbf{x_0} + \boldsymbol{p}) - f(\mathbf{x_0}) = \frac{1}{2}\boldsymbol{p}^T\nabla^2 f(\boldsymbol{\psi})\boldsymbol{p}$. For $\mathbf{x_0}$ to be a strict minimum the left side must be positive for every small $\boldsymbol{p} \ne 0$, so the quadratic form must be positive.