Calculus/Rolle's Theorem

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

Examples


Example:

$$ f(x) = x^2 - 3x $$. 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.

Solution:

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.