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.

In graphical terms, what this means is:

- 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) = x^{2}-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) = x

^{2}-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.

f'(x)=2x-5

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).

f(1)=1^{2}-5(1)+4=0

f(4)=4^{2}-5(4)+4=0

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.

0=2x-5

5=2x

x=2.5

The function’s slope is zero at x=2.5.

*That’s it!*

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