Suppose f is a given function having continuous second partial derivative. Let x* be a stationary point with positive semidefinite Hessian matrix. Which of the following statement is true?

Group of answer choices

x* can only be a local minimizer.
x* can only be a local maximizer.
x* can only be a saddle point.
x* can only be a local minimizer or a local maximizer.
x* can be a local minimizer, local maximizer, or saddle point

Given that \( x^* \) is a stationary point and the Hessian matrix at \( x^* \) is positive semidefinite, we can analyze the nature of the stationary point using the properties of the Hessian matrix.

### Key points:
1. **Stationary point**: At a stationary point \( x^* \), the gradient of the function \( f \) is zero, i.e., \( \nabla f(x^*) = 0 \).
2. **Positive semidefinite Hessian**: If the Hessian matrix \( H(x^*) \) is positive semidefinite, then for any vector \( v \), we have:
   \[
   v^T H(x^*) v \geq 0
   \]
   This means that the second-order directional derivatives are non-negative in every direction.

### Interpretation:
- A **positive semidefinite Hessian** suggests that the curvature is either positive or zero in all directions at \( x^* \).
- However, **positive semidefinite** does not imply strict positivity in all directions (which would be positive definite). It allows for the possibility that the Hessian has zero eigenvalues in some directions.
- Therefore, \( x^* \) could be a **local minimizer** if the Hessian is strictly positive in certain directions, but if there are directions where the Hessian is zero, it could also be a **saddle point**.
  
Thus, \( x^* \) can be:
- A **local minimizer**.
- A **saddle point**.
- It cannot be a **local maximizer**, because a positive semidefinite Hessian does not permit the necessary concavity for a local maximum.

### Conclusion:
The correct statement is:
- **\( x^* \) can be a local minimizer, local maximizer, or saddle point.**

This option corresponds to the following choice:
- **x\* can be a local minimizer, local maximizer, or saddle point.**

Would you like further details on how the Hessian helps classify stationary points?

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**5 Relative Questions:**
1. How does the definiteness of the Hessian affect the classification of stationary points?
2. What role does the second derivative test play in identifying local extrema?
3. Can a positive semidefinite Hessian guarantee convexity of the function?
4. How does a positive definite Hessian differ from a positive semidefinite Hessian in terms of classification of points?
5. What are the implications if the Hessian is negative semidefinite at a stationary point?

**Tip**: The eigenvalues of the Hessian matrix can provide important insights into the local curvature of the function and the nature of stationary points.

Explore how to classify stationary points using a positive semidefinite Hessian matrix. This problem covers multivariable calculus concepts, including stationary points, Hessian matrices, and the second derivative test. Suitable for undergraduate-level calculus students.

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