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Part CCSIR NET June 2023implicit-function-theorem-fails

Implicit function theorem fails

Consider the function f:R2→Rf : \mathbb{R}^{2} \to \mathbb{R} defined by f(x,y)=x2−y3f(x, y) = x^{2} - y^{3}. Which of the following statements are true?

  1. A.There is no continuous real-valued function g defined on any interval of R\mathbb{R} containing 0 such that f(x, g(x)) = 0.
  2. B.There is exactly one continuous real-valued function g defined on an interval of R\mathbb{R} containing 0 such that f(x, g(x)) = 0.✓ Correct
  3. C.There is exactly one differentiable real-valued function g defined on an interval of R\mathbb{R} containing 0 such that f(x, g(x)) = 0.
  4. D.There are two distinct differentiable real-valued functions g on an interval of R\mathbb{R} containing 0 such that f(x, g(x)) = 0.

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 hypothesis dropped.

See pricing

50 are analysed free — try those first.

The trap it tests

Hypothesis dropped

A theorem applied without checking one of its preconditions.

Drill statements like this

Related counterexample: A C¹ map with everywhere non-zero Jacobian is injective

More on this topic

The chapter behind this: Inverse and implicit function theorems — free to read

From Functions of Several Variables › Inverse and implicit function theorems, extrema

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