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GATE 2021 ME (ME2) – Question 37

Engineering Mathematics · Calculus: Partial derivatives, Taylor series, maxima and minima, Fourier series · 2 marks · Multiple choice

Let the superscript $T$ represent the transpose operation. Consider the function $f(x)=\frac12x^TQx-r^Tx$, where $x$ and $r$ are $n\times1$ vectors and $Q$ is a symmetric $n\times n$ matrix. The stationary point of $f(x)$ is

  1. $Q^Tr$
  2. $Q^{-1}r$
  3. $r/(r^Tr)$
  4. r

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Show answer and explanation

Correct answer: (B) $Q^{-1}r$

Explanation

The gradient of the symmetric quadratic is Qx; the linear term contributes−r. Stationarity therefore requires Qx=r.

When Q is nonsingular, $x=Q^{-1}r$, **B**. Symmetry alone does not guarantee nonsingularity: for singular Q there may be no stationary point or multiple points.