Worksheets · Calculus

Intermediate Value Theorem worksheet

A continuous function cannot get from one height to another without passing every height in between. The theorem says no more than that, so the two checks are always the same: is f continuous on the whole interval, and is the value really between f(a) and f(b)?

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Name

Intermediate Value Theorem

Date Period

Answer each question.

  1. Does the Intermediate Value Theorem guarantee a c in [−2,1] with f(c)=0 for f(x)=x−3?
  2. f is continuous on [−1,4]. What is the least number of zeros f must have on [−1,4]?x−101234f(x)5−4−5−8−12
  3. Find the value of c guaranteed by the Intermediate Value Theorem for f(x)=x2−3x−9 on [−3,0] with f(c)=−3.
  4. Does the Intermediate Value Theorem guarantee a c in [0,1] with f(c)=2 for f(x)=−x+3x+1?
  5. f is continuous on [−2,8]. What is the least number of zeros f must have on [−2,8]?x−202468f(x)6−36073
  6. Find the value of c guaranteed by the Intermediate Value Theorem for f(x)=x2+8x+14 on [−8,−5] with f(c)=2.
  7. Does the Intermediate Value Theorem guarantee a c in [−2,−1] with f(c)=0 for f(x)=−x3+2x−5?
  8. f is continuous on [−2,8]. What is the least number of zeros f must have on [−2,8]?x−202468f(x)−6−21−47−3
  9. Find the value of c guaranteed by the Intermediate Value Theorem for f(x)=x2+5x on [−5,−4] with f(c)=−1.
  10. Does the Intermediate Value Theorem guarantee a c in [3,6] with f(c)=−1 for f(x)=−x−5x−2?
  11. f is continuous on [0,5]. What is the least number of zeros f must have on [0,5]?x012345f(x)35238−8
  12. Find the value of c guaranteed by the Intermediate Value Theorem for f(x)=x2+5x−9 on [−8,−5] with f(c)=0.
  13. Does the Intermediate Value Theorem guarantee a c in [−3,1] with f(c)=1 for f(x)=x+3−2?
  14. f is continuous on [1,6]. What is the least number of zeros f must have on [1,6]?x123456f(x)−3762−12
  15. Find the value of c guaranteed by the Intermediate Value Theorem for f(x)=x2+5x+11 on [−1,1] with f(c)=8.
  16. Does the Intermediate Value Theorem guarantee a c in [−5,0] with f(c)=10 for f(x)=4x+1x+2?
  17. f is continuous on [−1,9]. What is the least number of zeros f must have on [−1,9]?x−113579f(x)−1618−2−2
  18. Find the value of c guaranteed by the Intermediate Value Theorem for f(x)=x2−14x+39 on [3,6] with f(c)=−1.
  19. Does the Intermediate Value Theorem guarantee a c in [−1,7] with f(c)=−5 for f(x)=x+2−3?
  20. f is continuous on [−2,8]. What is the least number of zeros f must have on [−2,8]?x−202468f(x)02−342−3

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

Sigma Prep · sigmaprep.io/worksheets

Name

Intermediate Value TheoremAnswer keyVersion 1

Date Period

Answer each question.

  1. Does the Intermediate Value Theorem guarantee a c in [−2,1] with f(c)=0 for f(x)=x−3?Not guaranteed: f is not continuous on [−2,1]
  2. f is continuous on [−1,4]. What is the least number of zeros f must have on [−1,4]?x−101234f(x)5−4−5−8−122
  3. Find the value of c guaranteed by the Intermediate Value Theorem for f(x)=x2−3x−9 on [−3,0] with f(c)=−3.c=3−332
  4. Does the Intermediate Value Theorem guarantee a c in [0,1] with f(c)=2 for f(x)=−x+3x+1?Yes: f(0)=3 and f(1)=1
  5. f is continuous on [−2,8]. What is the least number of zeros f must have on [−2,8]?x−202468f(x)6−360733
  6. Find the value of c guaranteed by the Intermediate Value Theorem for f(x)=x2+8x+14 on [−8,−5] with f(c)=2.c=−6
  7. Does the Intermediate Value Theorem guarantee a c in [−2,−1] with f(c)=0 for f(x)=−x3+2x−5?Not guaranteed: 0 is not between f(−2)=−1 and f(−1)=−6
  8. f is continuous on [−2,8]. What is the least number of zeros f must have on [−2,8]?x−202468f(x)−6−21−47−34
  9. Find the value of c guaranteed by the Intermediate Value Theorem for f(x)=x2+5x on [−5,−4] with f(c)=−1.c=−5−212
  10. Does the Intermediate Value Theorem guarantee a c in [3,6] with f(c)=−1 for f(x)=−x−5x−2?Not guaranteed: −1 is not between f(3)=−8 and f(6)=−114
  11. f is continuous on [0,5]. What is the least number of zeros f must have on [0,5]?x012345f(x)35238−81
  12. Find the value of c guaranteed by the Intermediate Value Theorem for f(x)=x2+5x−9 on [−8,−5] with f(c)=0.c=−5−612
  13. Does the Intermediate Value Theorem guarantee a c in [−3,1] with f(c)=1 for f(x)=x+3−2?Not guaranteed: 1 is not between f(−3)=−2 and f(1)=0
  14. f is continuous on [1,6]. What is the least number of zeros f must have on [1,6]?x123456f(x)−3762−123
  15. Find the value of c guaranteed by the Intermediate Value Theorem for f(x)=x2+5x+11 on [−1,1] with f(c)=8.c=−5+132
  16. Does the Intermediate Value Theorem guarantee a c in [−5,0] with f(c)=10 for f(x)=4x+1x+2?Not guaranteed: f is not continuous on [−5,0]
  17. f is continuous on [−1,9]. What is the least number of zeros f must have on [−1,9]?x−113579f(x)−1618−2−22
  18. Find the value of c guaranteed by the Intermediate Value Theorem for f(x)=x2−14x+39 on [3,6] with f(c)=−1.c=4
  19. Does the Intermediate Value Theorem guarantee a c in [−1,7] with f(c)=−5 for f(x)=x+2−3?Not guaranteed: −5 is not between f(−1)=−2 and f(7)=0
  20. f is continuous on [−2,8]. What is the least number of zeros f must have on [−2,8]?x−202468f(x)02−342−34

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

How to do these

Does the IVT guarantee a c in [1,3] with f(c)=5 for f(x)=x2−2x+3?

  1. f is a polynomial, so it is continuous on [1, 3].
  2. f(1) = 2 and f(3) = 6.
  3. 5 is between 2 and 6, so the theorem guarantees a c in [1, 3] with f(c) = 5.

The answer is Yes: f(1)=2 and f(3)=6.

Where students go wrong

Skipping the continuity check. 1/x is -1 at x = -1 and 1 at x = 1, but it never equals 0: it jumps over it at the asymptote, and the theorem says nothing.

Also called IVT, intermediate value property or Bolzano's theorem.

What you can put on this worksheet

Guaranteed or not: continuity and the value in between
Does the Intermediate Value Theorem guarantee a c in [−1,1] with f(c)=7 for f(x)=x2+3x+1? → Not guaranteed: 7 is not between f(−1)=−1 and f(1)=5
The Intermediate Value Theorem
Does the Intermediate Value Theorem guarantee that f(x)=−x3+x+4 has a zero on [−3,−2]? → No: f(−3)=28 and f(−2)=10 have the same sign
The fewest zeros a table forces
f is continuous on [0,4]. What is the least number of zeros f must have on [0,4]?x01234f(x)8−72−12 → 4
Find the c the theorem promises
Find the value of c guaranteed by the Intermediate Value Theorem for f(x)=x2+4x−8 on [−2,2] with f(c)=−3. → c=1

Questions about these worksheets

Yes. Every intermediate value 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 intermediate value 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 4: guaranteed or not: continuity and the value in between, the intermediate value theorem, the fewest zeros a table forces and find the c the theorem promises. Tick as many as you want and set how many of each, or let it spread them evenly.

Skipping the continuity check. 1/x is -1 at x = -1 and 1 at x = 1, but it never equals 0: it jumps over it at the asymptote, and the theorem says nothing.