Statistics How To

Rolle’s Theorem: Definition and Calculating

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.

rolle'stheorem-150x150
In graphical terms, what this means is:

  1. Take any interval on the x-axis (for example, -10 to 10). Make sure two of your function values are equal.
  2. 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]

The function f(x) = x2-5x+4 [1,4]. Graph generated with this HRW graphing calculator.




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.
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)=12-5(1)+4=0
f(4)=42-5(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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Rolle’s Theorem: Definition and Calculating was last modified: October 12th, 2017 by Stephanie Glen

5 thoughts on “Rolle’s Theorem: Definition and Calculating

  1. P.V.Raja Chowdary

    Sir,I agree that if f(a)=f(b),there will be atleast one local maxima and local minima at point x=’c’ at which the slope or derivative is 0.But even if f(a) is not equal to f(b) ,say f(b) lies above f(a),while the function f(x) is continuous,there exist a point at x=c,where the tangent still geometrically appears to parallel to x-axis i.e slope or derivative is “0”,then what is use of the condition,”f(a)=f(b)”?

  2. Andale Post author

    Assuming f(a)=f(b) isn’t true. You could draw a straight line from one side of the paper to the other and never have a point where the tangent is parallel. For example, a graph of x^2 from (0,100) does not have a tangent line of zero.

  3. Belle

    could you please draw a graph for the function f(x) = x2-5x+4 [1,4]. im confused. and if theres any link on how to draw graphs for function please help me to it. i appreciate your detailed explanation by the way. it has really helped me understand what the Rolle’s theorem and its process go about.