If f ( x )= k x +2 x x + x is strictly increasing for all real values of x , then

Step-by-step solution

For f(x) = k x + 2 x x + x to be strictly increasing, its derivative must be positive for all x where defined. Differentiating using the quotient rule, let u = k x + 2 x with u' = k x - 2 x , and v = x + x with v' = x - x . Then: f'(x) = (k x - 2 x)( x + x) - (k x + 2 x)( x - x) ( x + x)^2 Expanding and simplifying, the numerator reduces to k( ^2 x + ^2 x) - 2( ^2 x + ^2 x) = k - 2 . Hence, f'(x) = k - 2 ( x + x)^2 . The denominator is always positive when defined, so f'(x) > 0 only when k - 2 > 0 , i.e., k > 2 . T