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
                      y = √(1 - x²)

Product P = x y = x √(1-x²)

dP/dx = √(1-x²)  -  x * x / √(1-x²)
          = (1 - 2 x²) / √(1 - x²)   = 0      =>   x = + 1/√2  or  -1/√2
d²P/dx² = [ √(1-x²) (-4x) - (1- 2x²) ( -x/√(1-x²) ) ] / (1 - x²)
             =  [ -4 + 4 x³  +  x - 2 x³  ] / (1-x²)³/²
             =  [ 2 x³ + x - 4 ] / (1-x²)³/²
As   |x| < 1,  Numerator  is negative.  Hence P = xy is maximum at x = 1/√2

Answer:    x = 1/√2 ,   y = 1/√2       or  x = -1/√2  , y = -1/√2

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Simpler way:

Let     x = cosФ   y = sinФ 
Product =   xy = sinФ  cosФ = 1/2 * sin 2Ф
      It is maximum for Ф = π/4 or  5π/4
      So  x = y = 1/√2       or     x = y = -1/√2