Find the maximum and minimum values of f(x, y)=9x^2+10y^2

The function has no maximum since it is strictly increasing with any positive change of x or y. Finding the partial derivatives with respect to x and y give: f_x(x,y) = 18x f_y(x,y)= 20y Setting both equal to 0, gives us the coordinates of the only critical point (0,0). f(0,0) = 0. The minimum is therefore, 0.