Question

Consider the function \[ f(x, y) = x^2y + 2xy^2 - 2x^2y^2. \] Then which one of the following statements is correct?

• A. \( \left( \frac{3}{2}, 0 \right) \) is a point of local maxima of \( f \).
• B. \( \left( 0, \frac{3}{4} \right) \) is a point of local minima of \( f \).
• C. \( \left( \frac{3}{2}, \frac{3}{4} \right) \) is a point of local maxima of \( f \).
• D. \( \left( \frac{3}{2}, \frac{3}{4} \right) \) is a saddle point of \( f \).

Hint:

To determine the nature of a critical point, compute the second-order partial derivatives and evaluate the discriminant \( D \). If \( D<0 \), the point is a saddle point.

Answer 1

The Correct Option is D

To determine the nature of the critical points, we first compute the first-order partial derivatives of \( f(x, y) \): \[ f_x = 2xy + 2y^2 - 4xy^2, f_y = x^2 + 4xy - 4x^2y. \] Next, we set both \( f_x = 0 \) and \( f_y = 0 \) to find the critical points. After solving the system of equations, we find the critical point at \( \left( \frac{3}{2}, \frac{3}{4} \right) \). 
Now, we calculate the second-order partial derivatives: \[ f_{xx} = 2y - 4y^2, f_{yy} = 4x - 4x^2, f_{xy} = 4y - 8xy. \] At the critical point \( \left( \frac{3}{2}, \frac{3}{4} \right) \), we compute the discriminant \( D = f_{xx} f_{yy} - (f_{xy})^2 \). After evaluating, we find that \( D<0 \), which indicates that the critical point is a saddle point. Thus, \( \left( \frac{3}{2}, \frac{3}{4} \right) \) is a saddle point of \( f \).