Calculus/Rolle's Theorem

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Rolle's Theorem
Rolle's Theorem
If a function,  f(x) \ , is continuous on the closed interval  [a,b] \ , is differentiable on the open interval  (a,b) \ , and  f(a) = f(b) \ , then there exists at least one number c, in the interval  (a,b) \ such that  f'(c) = 0 \ .

Rolle's Theorem is important in proving the Mean Value Theorem.


Rolle's theorem.svg


 f(x) = x^2 - 3x . Show that Rolle's Theorem holds true somewhere within this function. To do so, evaluate the x-intercepts and use those points as your interval.


1: The question wishes for us to use the x-intercepts as the endpoints of our interval.

Factor the expression to obtain x(x-3)= 0 . x = 0 and x = 3 are our two endpoints. We know that f(0) and f(3) are the same, thus that satisfies the first part of Rolle's theorem (f(a) = f(b)).

2: Now by Rolle's Theorem, we know that somewhere between these points, the slope will be zero. Where? Easy: Take the derivative.

 dy \over dx  = 2x - 3

Thus, at  x = 3/2 , we have a spot with a slope of zero. We know that 3/2 (or 1.5) is between 0 and 3. Thus, Rolle's Theorem is true for this (as it is for all cases). This was merely a demonstration.