Calculus > Rolle’s Theorem
What is Rolle’s Theorem?
Rolle’s theorem is a special case of the mean value theorem. Rolle’s theorem states that for any continuous, differentiable function that has two equal values at two distinct points, the function must have a point on the function where the first derivative is zero. The technical way to state this is: if f is continuous and differentiable on a closed interval [a,b] and if f(a)=f(b), then f has a minimum of one value c in the open interval [a,b] so that f'(c)=0.
- Take any interval on the x-axis (for example, -10 to 10). Make sure two of your function values are equal.
- Draw a line from the beginning of the interval to the end. It doesn’t matter if the line is curved, straight or a squiggle — somewhere along that line you’re going to have a horizontal tangent line where the derivative, (f’) is zero. Try it!
How to use Rolle’s Theorem
Sample question: Use Rolle’s theorem for the following function: f(x) = x2-5x+4 for x-values [1,4]
Step 1: Determine if the function is continuous. You can only use Rolle’s theorem for continuous functions. This function f(x) = x2-5x+4 is a polynomial and is therefore continuous for all values of x. (How to check for continuity of a function).
Step 2: Figure out if the function is differentiable. If it isn’t differentiable, you can’t use Rolle’s theorem. the easiest way to figure out if the function is differentiable is to take the derivative.
Step 3: Check that the derivative is continuous, using the same rules you used for Step 1. f'(x)=2x-5 is a continuous function. If the derivative function isn’t continuous, you can’t use Rolle’s theorem.
Step 4: Plug the given x-values into the given formula to check that the two points are the same height (if they aren’t, then Rolle’s does not apply).
Both points f(1) and f(4) are the same height, so Rolle’s applies.
Step 5: Set the first derivative formula (from Step 2) to zero in order to find out where the function’s slope is zero.
The function’s slope is zero at x=2.5.
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