Let f(x) = ( ⁻¹ x)^2 + ( ⁻¹ x)^2 be a real-valued function defined on its domain. Then the sum of the greatest and the least values of f(x) is

Step-by-step solution

Given f(x) = ( ⁻¹ x)^2 + ( ⁻¹ x)^2 with domain x [-1, 1] . Using the identity ⁻¹ x + ⁻¹ x = 2 , we can write: f(x) = ( ⁻¹ x)^2 + ( 2 - ⁻¹ x )^2 Let ⁻¹ x = t , where t [- 2 , 2 ] . f(t) = t^2 + ( 2 - t )^2 = 2t^2 - t + ^2 4 Completing the square, we get: f(t) = 2 (t^2 - 2 t ) + ^2 4 = 2 (t - 4 )^2 - 2 ( ^2 16 ) + ^2 4 f(t) = 2 (t - 4 )^2 + ^2 8 The minimum value of f(t) occurs at t = 4 , which lies in the interval [- 2 , 2 ] . Minimum value = ^2 8 The maximum value of f(t) occurs at the endpoint furthest from t = 4