Find the maximum value of f(x,y) = x^2y^7 for x,y ≥ 0 on the unit circle

Question

Find the maximum value of f(x,y) = x2y7 for x,y ≥ 0 on the unit circle

Answer 1

Lagrange multiplier method.

f(x) = x²y⁷           f_x = 2xy⁷   f_y = 7x²y⁶

g(x,y) = x²+y² - 1     g_x = 2x     g_y = 2y

 

f_x - λg_x = 0                                                          f_y - λg_y = 0  

2xy⁷ - λ(2x) = 0 leads to x=0 or λ=y⁷                    the above leads to y⁶(7x²-2y²) = 0 and eventually to

                                                                       y=0 or 7x² = 2(1-x²) and thus x = √2 / 3 so y=√7 / 3.

 

In the cases where x or y = 0, then z = 0; in the case where (x,y) have the above radical values, z has its maximum,

z = (2/9)(√7 / 3)⁷ which you may evaluate decimally if desired.