f(x,y)=xx2+y2/ x for x ≠ 0 and 0 for x = 0. Which statements hold at (0,0)?

Both partial derivatives equal 1.

GATE 2026 MA – Question 22

Calculus · Multivariable Differentiation and Optimization · 1 mark · Multiple select

$f(x,y)=x\sqrt{x^2+y^2}/|x|$ for x ≠ 0 and 0 for x = 0. Which statements hold at (0,0)?

f is continuous.
Every directional derivative exists.
Both partial derivatives equal 1.
f is differentiable.

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Correct answer: (A) f is continuous.; (B) Every directional derivative exists.

Explanation

Bound |f(x,y)| ≤ √(x²+y²), so A holds. Along u = (a,b), the two-sided directional quotient is sign(a)√(a²+b²) if a ≠ 0 and zero if a = 0, so B holds. The coordinate partials are 1 and 0, refuting C. A derivative would be L(x,y) = x, but along y = x > 0 the error f−x = (√2−1)x is not negligible relative to √2x. Therefore D fails. **A,B**.

GATE 2026 MA – Question 22

Explanation

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