Calculus/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.

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.