Question

Consider the function \( F: \mathbb{R}^2 \to \mathbb{R}^2 \) given by \[ F(x, y) = (x^3 - 3xy^2 - 3x, 3x^2y - y^3 - 3y). \] Then, for the function \( F \), the inverse function theorem is:

• A. applicable at all points of \( \mathbb{R}^2 \)
• B. not applicable at exactly one point of \( \mathbb{R}^2 \)
• C. not applicable at exactly two points of \( \mathbb{R}^2 \)
• D. not applicable at exactly three points of \( \mathbb{R}^2 \)

Hint:

The inverse function theorem fails where the Jacobian matrix has a zero determinant. Find where the determinant vanishes to determine the points where the theorem does not apply.

Answer 1

The Correct Option is C

To apply the inverse function theorem, we need to check the determinant of the Jacobian matrix of \( F \). The inverse function theorem fails where the Jacobian determinant is zero.

The Jacobian matrix of \( F \) is:
\[ J_F(x, y) = \begin{pmatrix} \frac{\partial F_1}{\partial x} & \frac{\partial F_1}{\partial y} \\ \frac{\partial F_2}{\partial x} & \frac{\partial F_2}{\partial y} \end{pmatrix} = \begin{pmatrix} 3x^2 - 3y^2 - 3 & -6xy \\ 6xy - 3y^2 & 3x^2 - 3 \end{pmatrix} \]

We compute the determinant of \( J_F(x, y) \):
\[ \det(J_F(x, y)) = (3x^2 - 3y^2 - 3)(3x^2 - 3) - (-6xy)(6xy - 3y^2) \]

By simplifying the determinant expression, we find that the inverse function theorem fails at exactly two points. Therefore, the correct answer is:

\[ \boxed{\text{(C) not applicable at exactly two points of } \mathbb{R}^2.} \]