To determine where the inverse function theorem is applicable, we need to analyze the Jacobian matrix of the function \( F: \mathbb{R}^2 \to \mathbb{R}^2 \). The inverse function theorem can be applied at a point if the determinant of the Jacobian at that point is nonzero.
The function given is:
\( F(x, y) = (x^3 - 3xy^2 - 3x, 3x^2y - y^3 - 3y) \).
We will compute the Jacobian matrix \( J \) of \( F \), which is the matrix of first-order partial derivatives:
\[ J = \begin{bmatrix} \frac{\partial F_1}{\partial x} & \frac{\partial F_1}{\partial y} \\ \frac{\partial F_2}{\partial x} & \frac{\partial F_2}{\partial y} \end{bmatrix} \]where \( F_1(x, y) = x^3 - 3xy^2 - 3x \) and \( F_2(x, y) = 3x^2y - y^3 - 3y \).
Calculating the partial derivatives, we have:
Thus, the Jacobian matrix is:
\[ J = \begin{bmatrix} 3x^2 - 3y^2 - 3 & -6xy \\ 6xy & 3x^2 - 3y^2 - 3 \end{bmatrix} \]The determinant of the Jacobian matrix can be calculated as:
\[ \text{det}(J) = (3x^2 - 3y^2 - 3)(3x^2 - 3y^2 - 3) - (-6xy)(6xy) \] \[ = (3x^2 - 3y^2 - 3)^2 - (-6xy)^2 \] \[ = (3x^2 - 3y^2 - 3)^2 - 36x^2y^2 \]Setting \(\text{det}(J) = 0\) will give us the points where the inverse function theorem is not applicable:
\[ (3x^2 - 3y^2 - 3)^2 = 36x^2y^2 \]This implies:
\[ 3x^2 - 3y^2 - 3 = \pm 6xy \]Solving for these conditions involves finding points \((x, y)\) on \(\mathbb{R}^2\) where these equalities hold.
Analyzing these equations, you get two different solutions, leading to two specific points where \(\text{det}(J) = 0\), making the inverse function theorem not applicable. Hence the correct choice is:
Let \( X = \{ f \in C[0,1] : f(0) = 0 = f(1) \} \) with the norm \( \|f\|_\infty = \sup_{0 \leq t \leq 1} |f(t)| \), where \( C[0,1] \) is the space of all real-valued continuous functions on \( [0,1] \).
Let \( Y = C[0,1] \) with the norm \( \|f\|_2 = \left( \int_0^1 |f(t)|^2 \, dt \right)^{\frac{1}{2}} \). Let \( U_X \) and \( U_Y \) be the closed unit balls in \( X \) and \( Y \) centered at the origin, respectively. Consider \( T: X \to \mathbb{R} \) and \( S: Y \to \mathbb{R} \) given by
\[ T(f) = \int_0^1 f(t) \, dt \quad \text{and} \quad S(f) = \int_0^1 f(t) \, dt. \]
Consider the following statements:
S1: \( \sup |T(f)| \) is attained at a point of \( U_X \).
S2: \( \sup |S(f)| \) is attained at a point of \( U_Y \).
Then, which one of the following is correct?
Let \( g(x, y) = f(x, y)e^{2x + 3y} \) be defined in \( \mathbb{R}^2 \), where \( f(x, y) \) is a continuously differentiable non-zero homogeneous function of degree 4. Then,
\[ x \frac{\partial g}{\partial x} + y \frac{\partial g}{\partial y} = 0 \text{ holds for} \]