High School
SAT
ACT
University
MCAT
LSAT
DAT
Wize University Calculus 1 Textbook > Limits

The Intermediate Value Theorem

ContinuityPrevious Section
The Epsilon Delta Definition of a LimitNext Section
Popular Courses
Find My Course
0:00 / 0:00

The Intermediate Value Theorem

Continuity is very powerful since it can tell us if a function attains certain values. In particular, continuity tells us function outputs between certain values.


Intermediate Value Theorem (IVT)

If f(x)f(x) is continuous on [a,b],[a, b], then for any LL between f(a)f(a)and f(b)f(b) there exists a point c∈(a,b)c\in(a, b) for which f(c)=L.f(c)=L.




Wize Concept
In other words, since ff is continuous, f(x)f(x) , covers all the values between f(a)f(a) and f(b)f(b).



Wize Tip
Whenever the question asks about roots or solutions of an equation L=0L=0 in IVT.


Procedure for Problem Solving

  1. Determine the domain of significance.
  2. Evaluate the function at the endpoints of the domain.
  3. Determine if the points of interest are within the end points. If not, consider choosing a new domain.
0:00 / 0:00

Example: Intermediate Value Theorem


Determine if the functionf(x)=x3−6x+4f(x)=x^3-6x+4has a root in the domain [0,1].[0,1].



f(0)=0−0+4=4f(1)=1−6+4=−1}f(1)<0<f(0)\left.\begin{array}{r} f(0) = 0-0+4 = 4\\ f(1) = 1-6+4 = -1 \end{array}\right\} f(1)<0<f(0)
Therefore, f(c)=0f(c)=0 for some c, 0<c<1,0 < c < 1, so at least one root exists as f(x) is continuous on [0,1].

checklist
Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
  2. When you're done, check the "I have answered this question" box below.
  3. View the solution and report whether you got it right or wrong.
Use the Intermediate Value Theorem to show that x2+x−3=2\sqrt{x^2+x-3}=2 has a solution on the interval (2,3).
Extra Practice
The Intermediate Value Theorem
Show that there is at least one solution to the equation ln⁡x=esin⁡(πx2)\ln x=e^{\sin\left(\frac{\pi x}{2}\right)}.
The Intermediate Value Theorem
Use the Intermediate Value Theorem to show that the function f(x)=x3−4x+2f\left(x\right)=x^3-4x+2 crosses the x−x-axis in the interval [0,2][0,2] .
In intermediate Value Theorem
Does 2x3−2+cos⁡(x)=02x^3-2+\cos(x)=0 have a root in (0,1) ?
In intermediate Value Theorem
Does 2x3−2+cos⁡(x)=02x^3-2+\cos(x)=0 have a root in (0,1) ?
The Intermediate Value Theorem: At Least One Solution
Show that the function f(x)=x3−7x2+6x−13=0f(x)=x^3-7x^2+6x-13=0 must possess at least one real-valued solution.
In intermediate Value Theorem
Does 2x3−2+cos⁡(x)=02x^3-2+\cos(x)=0 have a root in (0,1) ?
Intermediate Value Theorem
If f(x)f\left(x\right) is a continuous function on [−1,1]\left[-1,1\right], such that f(−1)=−πf\left(-1\right)=-\pi and f(1)=πf\left(1\right)=\pi, which statement is always true?
The Intermediate Value Theorem
Show that there is at least one solution to the equation ln⁡x=esin⁡(πx2)\ln x=e^{\sin\left(\frac{\pi x}{2}\right)}.
Indetermiante Value Theorem
Does x3−1+sin⁡(x)=0x^3-1+\sin\left(x\right)=0 have a root in [0,1]\left[0,1\right]?
The Intermediate Value Theorem
Show that the equation sin⁡x=x2−x−1\sin x=x^2-x-1 has at least one root in the interval (0,3)(0,3), i.e., there exists a number c∈(0,3)c \in (0,3) such that sin⁡c=c2−c−1\sin c=c^2-c-1 .
The Intermediate Value Theorem
Use the Intermediate Value Theorem to show that the function f(x)=x3−4x+2f\left(x\right)=x^3-4x+2 crosses the x−x-axis in the interval [0,2][0,2] .
The Intermediate Value Theorem
Determine if the function f(x)=x3−6x+4f(x)=x^3-6x+4 has a root in the domain [0,1] [0,1] .
Does 3x2+cos⁡x=33x^2+\cos{x}=3 have a root in (0,1)(0,1)?
In intermediate Value Theorem
Does 2x3−2+cos⁡(x)=02x^3-2+\cos(x)=0 have a root in (0,1) ?
The Intermediate Value Theorem: At Least One Solution
Show that the function f(x)=x3−7x2+6x−13=0f(x)=x^3-7x^2+6x-13=0 must possess at least one real-valued solution.
Practice: IVT & Equation Solutions
Practice: IVT & Equation Solutions
Show that there is at least one solution to the equation ln⁡x=esin⁡(πx2)\ln x=e^{\sin\left(\frac{\pi x}{2}\right)}.
The Intermediate Value Theorem
Let f(x)f(x) be a function so that
f(x),f′(x),f′′(x)f(x), f'(x), f''(x) exist and are continuous for all xx, and
∣f(x)−sin⁡(x)∣≤1/4|f(x) - \sin(x)| \leq 1/4 for all xx.
Continuity
Let f(x)f(x) be a continuous function on the open interval (a,b)(a, b). Which of the following four statements are always true.
Select all that apply.
Alex is thinking about tan⁡x\tan x function. He knows that tan⁡1>0\tan1>0 and tan⁡2<0\tan2<0 He concludes that tan⁡x\tan x has a root in (1, 2). Justify if he is right or wrong.
The intermediate value theorem
f(x)=15x5+43x3+3x−2f(x)=\frac{1}{5}x^5+\frac{4}{3}x^3+3x-2 has
f(x):[0,1]→[1,3]f(x):[0,1]\rightarrow[1,3] is continuous. Does f(x)f(x) intersect with y=2x+1y=2x+1 ?
final114
Megan was thinking about the function f(x)=3(x−3)3f(x)=\dfrac{3}{(x-3)^3} and she thought f(0)<0f(0)<0 and also f(4)>0f(4)>0 .So, by the Intermediate Value Theorem, this function has a root in the interval (0,4)(0,4). Is she right?
Indetermiante Value Theorem
Does x3−1+sin⁡(x)=0x^3-1+\sin\left(x\right)=0 have a root in [0,1]\left[0,1\right]?
The Intermediate Value Theorem
Determine if the function f(x)=x3−6x+4f(x)=x^3-6x+4 has a root in the domain [0,1] [0,1] .
Does 3x2+cos⁡x=33x^2+\cos{x}=3 have a root in (0,1)(0,1)?
In intermediate Value Theorem
Does 2x3−2+cos⁡(x)=02x^3-2+\cos(x)=0 have a root in (0,1) ?
The Intermediate Value Theorem: At Least One Solution
Show that the function f(x)=x3−7x2+6x−13=0f(x)=x^3-7x^2+6x-13=0 must possess at least one real-valued solution.
The Intermediate Value Theorem
Use the Intermediate Value Theorem to show that the function f(x)=x3−4x+2f\left(x\right)=x^3-4x+2 crosses the x−x-axis in the interval [0,2][0,2] .
The Intermediate Value Theorem
Show that the equation sin⁡x=x2−x−1\sin x=x^2-x-1 has at least one root in the interval (0,3)(0,3), i.e., there exists a number c∈(0,3)c \in (0,3) such that sin⁡c=c2−c−1\sin c=c^2-c-1 .
🦊 TRICKY! Megan Mah was thinking about the y=tan⁡xy=\tan x function. She thought tan (1) > 0 and also tan(2) < 0, so, by the Intermediate Value Theorem, this function has a root in the interval (1,2). Is she right?
Does x3−1+sin⁡(x)=0x^3-1+\sin\left(x\right)=0 have at least one root in [0,1]\left[0,1\right]?
Does the equation 9x+cos⁡ (3x)=89x+\cos\ (3x)=8 have a solution in the domain (0,1)?
Show that the equation sin⁡x=x2−x−1\sin x=x^2-x-1 has at least one root in the interval (0,3)(0,3), i.e., there exist a number c∈(0,3)c \in (0,3) such that sin⁡c=c2−c−1\sin c=c^2-c-1 . State the name of the theorem you use.
🦊 TRICKY! Use the Intermediate Value Theorem to show that the function f(x)=x3−4x+2f\left(x\right)=x^3-4x+2 crosses the x-axis at least twice in the interval [0,2].
Use Intermediate Value Theorem to conclude that x2+x−3=2\sqrt{x^2+x-3}=2 has at least one solution in the interval x∈(2,3)x\in(2,3).
Does the equation 3x2+cos⁡x=33x^2+\cos{x}=3 have a solution in the domain (0,1)(0,1)?
Does the equation 2x3−2+cos⁡(x)=02x^3-2+\cos(x)=0 have at least one solution in the domain (0,1)?
Determine if the polynomial x3−6xx^3-6x will attain the value of -4 in the domain [0,1].
Intermediate Value Theorem
Suppose f is a continuous function and f(0)=0 and f(1)=2. Is there a constant c∈(0,1)c\in (0,1) such that f(c)=2−c5f(c)=\sqrt[5]{2-c}? HINT: Apply the Intermediate Value Theorem on the function g(x)=[f(x)]5+xg(x)=[f(x)]^5+x.
The Intermediate Value Theorem
Show that there is at least one solution to the equation ln⁡x=esin⁡(πx2)\ln x=e^{\sin\left(\frac{\pi x}{2}\right)}.
Intermediate Value Theorem
If f(x)f\left(x\right) is a continuous function on [−1,1]\left[-1,1\right], such that f(−1)=−πf\left(-1\right)=-\pi and f(1)=πf\left(1\right)=\pi, which statement is always true?
Intermediate Value Theorem
Let f(x)f(x) be an even, continuous function and suppose that
f(−2019)=12f(-2019) = \frac{1}{2}
and that
Intermediate Value Theorem
Find an interval of length at most 1 that contains a solution to the equation
ex−2=0e^x - 2 = 0
Intermediate Value Theorem
Show that the function
p(x)=x4+x2−2p(x)=x^4+x^2-2
has a solution on the interval [0,2][0, 2] .
The Intermediate Value Theorem
Does 3x2+cos⁡x=33x^2+\cos x=3 has a root in (0, 1)?
f(x):[0,1]→[1,3]f(x):[0,1]\rightarrow[1,3] is continuous. Does f(x)f(x) intersect with y=2x+1y=2x+1 ?
Alex is thinking about tan⁡x\tan x function. He knows that tan⁡1>0\tan1>0 and tan⁡2<0\tan2<0 He concludes that tan⁡x\tan x has a root in (1, 2). Justify if he is right or wrong.