The function \( f(x, y)=x^{4}+y^{4}-2 x^{2}-18 y^{2}+20 \) has nine critical points, and among them are \( (1,3),(1,-3),(-1,0) \) and \( (-1,-3) \). Which of the following is correct about these critical points which are listed? a. Three are local minima and one is a saddle point b. Two are saddle points, one is a local maximum and one is a local minimum c. Three are saddle points and one is a local minimum d. Three are local minima and one is a local maximum e. Two are minima and two are saddle points

To determine the nature of the critical points for the function \( f(x, y)=x^{4}+y^{4}-2 x^{2}-18 y^{2}+20 \), we can analyze the second derivative test. Calculate the second derivatives to form the Hessian matrix and evaluate it at the critical points. For \( (1, 3) \) and \( (1, -3) \): The Hessian determinant is positive, indicating these are local minima. For \( (-1, 0) \): The Hessian determinant here is negative, indicating a saddle point. For \( (-1, -3) \): Similar to the first points, this point will also lead to a positive determinant indicating a local minimum. Therefore, the correct answer is: **e. Two are minima and two are saddle points.** The Hessian is essential in determining the behavior of critical points, so next time you have a function, don't forget about this powerful tool! The determinants relate closely to how the function curves around those points—almost like a rollercoaster ride for math lovers! If you want to delve deeper into the subject, I'd recommend checking out multivariable calculus textbooks or online courses dedicated to optimization techniques. They often have vivid examples that will make you appreciate the beauty of these mathematical concepts!