Rolle’s Theorem
If f(a) = f(b) Then f'(c) = 0 at some point check f(a) and f(b) if = then f'(x) = 0
Rolle’s Theorem states that if a function f(x) is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), and if f(a) = f(b), then there exists at least one value c in (a, b) such that f'(c) = 0.
In simpler terms, if a function is continuous and differentiable on an interval, and if its endpoints have the same function value, then at some point within the interval, the function’s derivative must be equal to zero.
This theorem is important in calculus as it allows us to determine if there are any points of horizontal tangency or flat spots along the curve of the function. We can also use Rolle’s Theorem to prove the existence of roots for certain types of equations.
For example, if we have a polynomial function f(x) such that f(a) = 0 and f(b) = 0, then by Rolle’s Theorem, there must be at least one value c in (a, b) where f'(c) = 0, which implies a critical point or a possible turning point. This can help us identify the location of the roots of the polynomial.
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