Question

Consider the function \( f(x, y) = x^3 - 3xy^2, x, y \in \mathbb{R} \). Which one of the following statements is TRUE?

• A. \( f \) has a local minimum at \( (0,0) \)
• B. \( f \) has a local maximum at \( (0,0) \)
• C. \( f \) has global maximum at \( (0,0) \)
• D. \( f \) has a saddle point at \( (0,0) \)

Hint:

In multivariable calculus, to identify critical points, compute the first and second derivatives and use tests to classify them. A saddle point is characterized by having both positive and negative curvatures along different directions.

Answer 1

The Correct Option is D

Step 1: Analyzing the function.
The function \( f(x, y) = x^3 - 3xy^2 \) is a multivariable polynomial. We can calculate the first and second partial derivatives to analyze its critical points. The function has a critical point at \( (0, 0) \). By examining the second derivative test or inspecting the function's behavior near this point, we find that \( (0,0) \) is a saddle point.
Step 2: Analyzing the options.
(A) \( f \) has a local minimum at \( (0,0) \): Incorrect. The function does not have a local minimum at \( (0,0) \).
(B) \( f \) has a local maximum at \( (0,0) \): Incorrect. The function does not have a local maximum at \( (0,0) \).
(C) \( f \) has global maximum at \( (0,0) \): Incorrect. The function does not have a global maximum at \( (0,0) \).
(D) \( f \) has a saddle point at \( (0,0) \): Correct. The function has a saddle point at \( (0,0) \), as the partial derivatives suggest that the critical point is neither a maximum nor a minimum.
Step 3: Conclusion.
The correct answer is (D) \( f \) has a saddle point at \( (0,0) \).