Consider the function f : ℝ² → ℝ defined by f(x, y) = x² − y³. Which of the following statements are true?

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

Implicit function theorem fails

Consider the function $f : R^{2} \to R$ defined by $f (x, y) = x^{2} - y^{3}$ . Which of the following statements are true?

A.There is no continuous real-valued function g defined on any interval of $R$ containing 0 such that f(x, g(x)) = 0.
B.There is exactly one continuous real-valued function g defined on an interval of $R$ containing 0 such that f(x, g(x)) = 0.✓
C.There is exactly one differentiable real-valued function g defined on an interval of $R$ containing 0 such that f(x, g(x)) = 0.
D.There are two distinct differentiable real-valued functions g on an interval of $R$ containing 0 such that f(x, g(x)) = 0.

Solution

$g (x)^{3} = x^{2}$ forces g(x) = $x^{2/3}$ (the real cube root is a bijection), so there is exactly one continuous solution. It is not differentiable at 0, so no differentiable g exists. The implicit function theorem does not apply since f_y(0,0) = 0.

The trap it tests

Hypothesis dropped

A theorem applied without checking one of its preconditions.

Drill statements like this

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

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