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Questions › DSE › DSE 2018 › Q26
4 marks, −1Mathematics › Multivariable calculus › Partial derivatives and differentiabilityHardabout 3 min

DSE 2018 Question 26: Multivariable calculus

Consider the function f(x,y)=xyx2+y2f(x,y) = \dfrac{xy}{x^2 + y^2}f(x,y)=x2+y2xy​ if (x,y)≠(0,0)(x,y) \neq (0,0)(x,y)=(0,0), and f(0,0)=0f(0,0) = 0f(0,0)=0. Which of the following is correct?

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Plain-text version

Consider the function f(x,y) = xy/(x² + y²) if (x,y) ≠ (0,0), and f(0,0) = 0. Which of the following is correct? (A) f is differentiable at (0,0) and its partial derivatives are not both 0 at (0,0) (B) f is not differentiable at (0,0) and its partial derivatives are both 0 at (0,0) (C) f is differentiable at (0,0) and its partial derivatives are both 0 at (0,0) (D) f is not differentiable at (0,0) and its partial derivatives do not exist at (0,0)

Also written as: Consider the function f(x,y) = xy/(x^2 + y^2) if (x,y) != (0,0), and f(0,0) = 0. Which of the following is correct?

Multivariable calculus has come up 32 times in 7 exams since 2007.

What this question tests

This question is on Partial derivatives and differentiability, part of Multivariable calculus in Mathematics.

Derivatives with respect to one variable at a time, and how they relate to continuity and differentiability.

TestStop has 12 past-paper questions on this subtopic; the nearest in year are below.

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Practise Multivariable calculusMore on Partial derivatives and differentiability

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Question from the DSE 2018 paper. Solutions are TestStop's own. Not affiliated with or endorsed by any examination authority.