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GATE 2021 MA – Question 47

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

Define f(x,y)=√(x²+y²) sin(y²/x) for x≠0 and zero for x=0. P:f is continuous but not differentiable at (0,0). Q:every unit-direction derivative exists there. Choose truth values.

  1. Both P and Q true
  2. P true, Q false
  3. P false, Q true
  4. Both false

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Show answer and explanation

Correct answer: (A) Both P and Q true

Explanation

The bound |f|≤√(x²+y²) proves continuity. On a line (x,y)=t(a,b), a≠0, the directional quotient equals √(a²+b²)(|t|/t)sin(tb²/a)→0; the vertical line also gives zero. Both partials are therefore zero. But along x=y², f/√(x²+y²)=sin1≠0, so the zero linear map is not a derivative. Both statements hold, **A**.

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