Part 6/7:
To investigate if multiple roots exist, we can apply Rolle's Theorem again under the assumption of two roots. If ( f(a) = 0 ) and ( f(b) = 0 ) for two distinct points ( a ) and ( b ), then there must exist a point ( c ) in between where ( f'(c) = 0 ). Calculating the derivative ( f'(x) = 3x^2 + 1 ), it is clear that this is always greater than zero, indicating that the function is strictly increasing. Thus, there cannot be multiple roots, confirming only one real root exists.