(b) Determine and classify the stationary points of the function: f(x,y)=x^3+y^2-5xy-5y+10 (15 marks)

To find the stationary points of the function , we need to find the points where the gradient of the function is equal to the zero vector. In other words, we need to find the points where the partial derivatives of the function with respect to x and y are both equal to zero.The gradient of the function is given by the vector . To find the stationary points, we need to set this gradient equal to the zero vector and solve for x and y. Setting this gradient equal to the zero vector, we get the following system of equations: Solving this system of equations, we find that the stationary points of the function are the points (5/3, -5/2) and (5/2, -5/2).To classify these stationary points, we need to compute the Hessian matrix of the function, which is given by the matrix .Evaluating the second partial derivatives, we get: Substituting the values of x and y from the stationary points, we can compute the eigenvalues of the Hessian matrix to classify the stationary points.For the stationary point (5/3, -5/2): The eigenvalues of this matrix are and . Since one of the eigenvalues is positive and the other is negative, this stationary point is a saddle point.For the stationary point (5/2, -5/2): The eigenvalues of this matrix are and . Since one of the eigenvalues is positive and the other is negative, this stationary point is also a saddle point.Therefore, the function has two saddle points at the points (5/3, -5/2) and (5/2, -5/2).