Question

Find the points ( x, y ) on the unit circle, at which the product xy is maximum or minimum.

Answer 1

unit circle:    x² + y² = 1    with center at O(0,0)

Differentiating the above eq. wrt x:   2 x + 2 y y' = 0
                                                           y' = - x/y
P = x y 
dP/dx = y + x y' = y - x² /y  = (y² - x²) / y
          or, (2 y² - 1 ) / y 
dP/dx = 0   =>    y = +x or -x    or    y = +1/√2  or  -1/√2

Hence points:
   xy  maximum at:   (1/√2,  1/√2)  , (-1/√2 , -1/√2)
   xy  minimum  at:   (1/√2, -1/√2),   (-1/√2 , 1/√2)