Part CCSIR NET December 2024every-line-behaves-and-a-parabola-does-not

Every line behaves and a parabola does not

Consider the function $f : \mathbb{R}^{2} \to \mathbb{R}$ defined by $f(x, y) = x^{2}y/(x^{4} + y^{2}) + e^($ xy) for (x, y) ≠ (0, 0), and f(x, y) = 1 for (x, y) = (0, 0). Which of the following statements are true?

A.f is differentiable on $\mathbb{R}^{2}$ \ {(0, 0)}.✓ Correct
B.All the directional derivatives of f exist at (0, 0).✓ Correct
C.f is differentiable on $\mathbb{R}^{2}$ .
D.f is not continuous at (0, 0).✓ Correct

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests standard counterexample.

See pricing

50 are analysed free — try those first.

The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

Drill statements like this

Every line behaves and a parabola does not

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