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GATE 2024 MA – Question 54

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

f(x,y)=(1−cos(x²/y²))√(x²+y²) for y≠0, and zero otherwise. Which statements hold at (0,0)?

  1. Continuous but not differentiable
  2. Differentiable
  3. All directional derivatives exist and are zero
  4. Both partial derivatives exist and are zero

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Correct answer: (A) Continuous but not differentiable; (D) Both partial derivatives exist and are zero

Explanation

Bound |f|≤2√(x²+y²) proves continuity. Both axes have f=0, giving partials zero. A derivative would therefore be zero, but on y=x≠0, $f/\sqrt{x^2+y^2}=1-\cos1\ne0$, refuting differentiability. The directional quotient along (1,1) is √2(1−cos1)|t|/t, so the two-sided derivative does not exist; even one-sided it is nonzero. **A,D**.

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