[5 marks] Locate all relative maxima, relative minima and saddle points of thefunction f (x, y) = xy − x3 − y2.

To find the relative maxima, relative minima and saddle points of the function f(x, y) = xy - x³ - y², we need to follow these steps:

Step 1: Find the first partial derivatives of the function with respect to x and y.

f_x = y - 3x² f_y = x - 2y

Step 2: Set these first partial derivatives equal to zero and solve for x and y to find the critical points.

y - 3x² = 0 x - 2y = 0

Solving these equations simultaneously gives us two critical points: (0,0) and (1,1/2).

Step 3: Find the second partial derivatives of the function.

f_xx = -6x f_yy = -2 f_xy = 1

Step 4: Evaluate the second partial derivatives at the critical points.

At (0,0): f_xx = 0, f_yy = -2, f_xy = 1 At (1,1/2): f_xx = -6, f_yy = -2, f_xy = 1

Step 5: Use the second derivative test to classify the critical points. The second derivative test uses the determinant of the Hessian matrix, D = f_xx * f_yy - (f_xy)².

At (0,0): D = 0 * -2 - (1)² = -1, which is less than 0, so (0,0) is a saddle point. At (1,1/2): D = -6 * -2 - (1)² = 12 - 1 = 11, which is greater than 0. Since f_xx < 0, (1,1/2) is a relative maximum.

So, the function f(x, y) = xy - x³ - y² has a saddle point at (0,0) and a relative maximum at (1,1/2). It does not have a relative minimum.