Final answer:
To find the local maximum and minimum values and saddle points of the function f(x, y) = x³ + y³ - 3x² - 9y² - 9x, we need to take the partial derivatives, set them equal to zero, and use the second partial derivatives test.
Step-by-step explanation:
To find the local maximum and minimum values and saddle points of the function f(x, y) = x³ + y³ - 3x² - 9y² - 9x, we need to take the partial derivatives with respect to x and y and set them equal to zero. The partial derivative with respect to x is 3x² - 6x - 9 and the partial derivative with respect to y is 3y² - 18y. Setting both of these equal to zero, we can solve for x and y to find the critical points. Then, we can use the second partial derivatives test to determine if these critical points are local maxima, local minima, or saddle points.
Calculating the second partial derivatives, we find that the second derivative with respect to x is 6x - 6 and the second derivative with respect to y is 6y - 18. Plugging in the critical points, we can evaluate these second partial derivatives to determine the behavior of the function at each critical point. Finally, we can identify the local maximum and minimum values and saddle points based on the results of the second partial derivatives test.