Final answer:
To find and classify critical points of the given functions, we calculate the gradient of each function, set it to zero, and solve for x and y. We then use the second derivative test to classify the critical points.
Step-by-step explanation:
The student has asked to find and classify all the critical points of two given functions, which is a topic in multivariable calculus. A critical point occurs where the gradient (the vector of partial derivatives) of the function is zero or undefined. To find the critical points, we set the partial derivatives of the functions with respect to x and y to zero and solve for x and y.
For function (a) f(x, y) = 3xy - 1/2y² + 2x³ + 9/2x², we would take the partial derivatives ∂f/∂x and ∂f/∂y, set them to zero, and solve the resulting system of equations. Similarly, for function (b) f(x, y) = x³ - 3x + 3y², we follow the same steps to find its critical points.
Once the critical points are found, we classify them using the second derivative test, which involves evaluating the Hessian matrix at those critical points. The sign of the determinant of the Hessian matrix and the signs of its entries will determine whether the critical points are local maxima, local minima, or saddle points.