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GATE 2023 MA – Question 52

Calculus · Multivariable Differentiation and Optimization · 2 marks · Multiple select

Let f(x,y)=2x²y/(x²+y²) away from the origin and f(0,0)=0. Which statements hold?

  1. Directional derivative in direction (1/√2,1/√2) equals 1/√2.
  2. Directional derivative in direction (0,1) equals 1.
  3. Directional derivative in direction (1,0) equals 0.
  4. f is not differentiable at (0,0).

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Show answer and explanation

Correct answer: (A) Directional derivative in direction (1/√2,1/√2) equals 1/√2.; (C) Directional derivative in direction (1,0) equals 0.; (D) f is not differentiable at (0,0).

Explanation

For a unit vector (a,b), f(ta,tb)/t=2a²b. This is 1/√2 in the diagonal direction and zero on both axes. The partial derivatives are zero, so differentiability would imply all directional derivatives zero, contradicting the diagonal value. **A,C,D**.

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