Worksheets · Calculus

Rolle's theorem worksheet

If a smooth curve comes back to the same height it started at, it has to turn round somewhere in between. That turning point is where the derivative is zero, and that is the whole theorem.

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Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

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Name

Rolle's Theorem

Date Period

Find each value of c, or say why the theorem does not apply.

  1. Find cc from Rolle's theorem

    f(x)=2x3−6x2−18x+60 on [−3,3]. Find the value of c guaranteed by Rolle's theorem.
  2. f(x)=−x3+12x2−45x+44 on [2,5]. Find the value of c guaranteed by Rolle's theorem.
  3. f(x)=−x3+3x2−1 on [−1,2]. Find the value of c guaranteed by Rolle's theorem.
  4. f(x)=2x3−24x2+72x−4 on [0,6]. Find the value of c guaranteed by Rolle's theorem.
  5. f(x)=x3−15x2+72x−110 on [3,6]. Find the value of c guaranteed by Rolle's theorem.
  6. f(x)=x3−9x2+15x+20 on [−1,5]. Find the value of c guaranteed by Rolle's theorem.
  7. f(x)=x3−21x2+135x−245 on [3,9]. Find the value of c guaranteed by Rolle's theorem.

  1. Check Rolle's theorem and find cc

    Does Rolle's theorem apply to f(x)=x2−6x on [0,6]? If so, find every c.
  2. Does Rolle's theorem apply to f(x)=3x−3 on [1,5]? If so, find every c.
  3. Does Rolle's theorem apply to f(x)=x2+2x−8 on [−4,2]? If so, find every c.
  4. Does Rolle's theorem apply to f(x)=5(x−2)2 on [1,3]? If so, find every c.
  5. Does Rolle's theorem apply to f(x)=(x−3)23 on [0,6]? If so, find every c.
  6. Does Rolle's theorem apply to f(x)=x+12x on [3,4]? If so, find every c.
  7. Does Rolle's theorem apply to f(x)=x2−4x+3 on [1,3]? If so, find every c.
  8. Does Rolle's theorem apply to f(x)=x3−x on [−1,1]? If so, find every c.
  9. Does Rolle's theorem apply to f(x)=1x−1 on [−1,3]? If so, find every c.
  10. Does Rolle's theorem apply to f(x)=x3+5x2+4x on [−4,0]? If so, find every c.
  11. Does Rolle's theorem apply to f(x)=∣x∣ on [−2,2]? If so, find every c.
  12. Does Rolle's theorem apply to f(x)=x3−3x2+2x on [0,2]? If so, find every c.
  13. Does Rolle's theorem apply to f(x)=(x+1)23 on [−4,2]? If so, find every c.

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Rolle's TheoremAnswer keyVersion 1

Date Period

Find each value of c, or say why the theorem does not apply.

  1. Find cc from Rolle's theorem

    f(x)=2x3−6x2−18x+60 on [−3,3]. Find the value of c guaranteed by Rolle's theorem.c=−1
  2. f(x)=−x3+12x2−45x+44 on [2,5]. Find the value of c guaranteed by Rolle's theorem.c=3
  3. f(x)=−x3+3x2−1 on [−1,2]. Find the value of c guaranteed by Rolle's theorem.c=0
  4. f(x)=2x3−24x2+72x−4 on [0,6]. Find the value of c guaranteed by Rolle's theorem.c=2
  5. f(x)=x3−15x2+72x−110 on [3,6]. Find the value of c guaranteed by Rolle's theorem.c=4
  6. f(x)=x3−9x2+15x+20 on [−1,5]. Find the value of c guaranteed by Rolle's theorem.c=1
  7. f(x)=x3−21x2+135x−245 on [3,9]. Find the value of c guaranteed by Rolle's theorem.c=5

  1. Check Rolle's theorem and find cc

    Does Rolle's theorem apply to f(x)=x2−6x on [0,6]? If so, find every c.Yes: c=3
  2. Does Rolle's theorem apply to f(x)=3x−3 on [1,5]? If so, find every c.No: f is not continuous at x=3
  3. Does Rolle's theorem apply to f(x)=x2+2x−8 on [−4,2]? If so, find every c.Yes: c=−1
  4. Does Rolle's theorem apply to f(x)=5(x−2)2 on [1,3]? If so, find every c.No: f is not continuous at x=2
  5. Does Rolle's theorem apply to f(x)=(x−3)23 on [0,6]? If so, find every c.No: f is not differentiable at x=3
  6. Does Rolle's theorem apply to f(x)=x+12x on [3,4]? If so, find every c.Yes: c=23
  7. Does Rolle's theorem apply to f(x)=x2−4x+3 on [1,3]? If so, find every c.Yes: c=2
  8. Does Rolle's theorem apply to f(x)=x3−x on [−1,1]? If so, find every c.Yes: c=−33 or c=33
  9. Does Rolle's theorem apply to f(x)=1x−1 on [−1,3]? If so, find every c.No: f is not continuous at x=1
  10. Does Rolle's theorem apply to f(x)=x3+5x2+4x on [−4,0]? If so, find every c.Yes: c=−5−133 or c=−5+133
  11. Does Rolle's theorem apply to f(x)=∣x∣ on [−2,2]? If so, find every c.No: f is not differentiable at x=0
  12. Does Rolle's theorem apply to f(x)=x3−3x2+2x on [0,2]? If so, find every c.Yes: c=3−33 or c=3+33
  13. Does Rolle's theorem apply to f(x)=(x+1)23 on [−4,2]? If so, find every c.No: f is not differentiable at x=−1

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

Find the value of c for f(x)=x2−4x on [0,4]

  1. Both ends are at zero, so the theorem applies.
  2. The derivative is 2x minus 4.
  3. That is zero at x equals 2.

The answer is c=2.

Where students go wrong

Checking only that the function is continuous. It also has to come back to the same height: if the two ends are at different heights there is no reason for a flat spot, and the mean value theorem is the one that applies instead.

Also called Rolle theorem, derivative equals zero or horizontal tangent on an interval.

What you can put on this worksheet

Rolle's theorem
f(x)=3x2+18x on [−6,0]. Find the value of c guaranteed by Rolle's theorem. → c=−3
Does it apply?
Does Rolle's theorem apply to f(x)=x3−x on [−1,1]? If so, find every c. → Yes: c=−33 or c=33
Two or more values of c, quotients, trig, and a pole
Does Rolle's theorem apply to f(x)=x2−6x on [0,6]? If so, find every c. → Yes: c=3

Questions about these worksheets

Yes. Every rolle's theorem sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the rolle's theorem questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 3: rolle's theorem, does it apply? and two or more values of c, quotients, trig, and a pole. Tick as many as you want and set how many of each, or let it spread them evenly.

Checking only that the function is continuous. It also has to come back to the same height: if the two ends are at different heights there is no reason for a flat spot, and the mean value theorem is the one that applies instead.