Question

Find the maximum and minimum value of x³ + y³ - 3axy.

Answer 1

Given, u = x³ + y³ - 3axy

Differentiating equation (i), we have

Now for maximum or minimum of u, we must have

But x = -a, y = a, do not satisfy equation (iii) hence are not solutions.

Hence, the solutions are x = 0, y = 0 and x = a , y = a

At x = 0 y = 0 we have : 

r = 0, s = - 3a, t = 0 

rt - s² = 0 - (- 3a)² = - ve 

and hence nor minimum at x = 0, y = 0 

At x = a, y = a, we have 

r = 6a, s = - 3a, t = 6a 

rt - s² = (6a)(6a) - (- 3a)² = 36a² - 9a² > 0 

Also, r = 6a > 0 if a > 0 and r < 0 if a < 0 

Hence, there is maximum or minimum according to a < 0 or a > 0.