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Part CCSIR NET December 2023monotone-not-necessarily-increasing

Monotone not necessarily increasing

For a differentiable surjective function f : (0,1) → (0,1), consider F:(0,1)2→(0,1)2,F(x,y)=(f(x),f(y))F : (0,1)^{2} \to (0,1)^{2}, F(x, y) = (f(x), f(y)). If f′(x) ≠ 0 for every x ∈ (0,1), which of the following statements are true?

  1. A.F is injective.✓ Correct
  2. B.f is increasing.
  3. C.For every (x′,y′)∈(0,1)2(x', y') \in (0,1)^{2}, there exists a unique (x,y)∈(0,1)2(x, y) \in (0,1)^{2} such that F(x, y) = (x′, y′).✓ Correct
  4. D.The total derivative DF(x, y) is invertible for all (x,y)∈(0,1)2(x, y) \in (0,1)^{2}.✓ 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 converse assumed.

See pricing

50 are analysed free — try those first.

The trap it tests

Converse assumed

The theorem runs one way. You used it in the other.

Drill statements like this

Related counterexample: If all partial derivatives exist at a point then f is continuous there

More on this topic

The chapter behind this: Differentiability in several variables — free to read

From Functions of Several Variables › Partial derivatives, differentiability, chain rule

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

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