If f(x) = (1 + x) - x 1+x , then the values of x for which f(x) is monotonically increasing and monotonically decreasing are respectively.....

Step-by-step solution

The given function is f(x) = (1 + x) - x 1+x The domain of the function is 1 + x > 0 x > -1 Differentiating f(x) with respect to x : f'(x) = 1 1+x - (1+x) 1 - x 1 (1+x)^2 f'(x) = 1 1+x - 1 (1+x)^2 f'(x) = 1+x-1 (1+x)^2 = x (1+x)^2 For f(x) to be monotonically increasing, f'(x) > 0 x (1+x)^2 > 0 x > 0 Thus, f(x) is monotonically increasing on (0, ) For f(x) to be monotonically decreasing, f'(x) x (1+x)^2 Considering the domain x > -1 , f(x) is monotonically decreasing on (-1, 0) The intervals for monotonically incre