Part 5/7:
Rolle's Theorem can be applied to real-world problems, such as determining the roots of a polynomial. Let’s examine the function ( f(x) = x^3 + x - 1 ) to show it has a single real root.
Evaluating the function at two specific points:
( f(0) = -1 ) (less than zero)
( f(1) = 1 ) (greater than zero)
By the Intermediate Value Theorem, this polynomial, being continuous, guarantees at least one root exists between ( 0 ) and ( 1 ).