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The Intermediate Value Theorem

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Wize University Calculus 1 Textbook > Limits

The Intermediate Value Theorem

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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.
Show that f(x)f(x) has at least one zero in the open interval (0,2π)(0, 2\pi)

More The Intermediate Value Theorem Questions:
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)}.
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.