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GATE 2022 EE – Question 50

Engineering Mathematics · Calculus: Directional derivatives · 2 marks · Multiple choice

Let, $f(x,y,z)=4x^2+7xy+3xz^2$. The direction in which the function $f(x,y,z)$ increases most rapidly at point $P=(1,0,2)$ is

  1. $20\hat\imath+7\hat\jmath$
  2. $20\hat\imath+7\hat\jmath+12\hat k$
  3. $20\hat\imath+12\hat k$
  4. $20\hat\imath$

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Correct answer: (B) $20\hat\imath+7\hat\jmath+12\hat k$

Explanation

The function increases fastest along its gradient: $\nabla f=(8x+7y+3z^2)\hat\imath+7x\hat\jmath+6xz\hat k$. At $(1,0,2)$ this is $20\hat\imath+7\hat\jmath+12\hat k$.