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

Monotone not necessarily increasing

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

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

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Monotone not necessarily increasing

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