Intermediate Value Theorem Math Example 2

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Example 2

medium
Use the IVT to show that cos⁡x=x\cos x = x has a solution in (0,1)(0, 1).

Solution

  1. 1
    Let g(x)=cos⁡x−xg(x) = \cos x - x. We want g(c)=0g(c) = 0 for some c∈(0,1)c \in (0, 1).
  2. 2
    gg is continuous on [0,1][0, 1] (sum of continuous functions).
  3. 3
    g(0)=cos⁡0−0=1>0g(0) = \cos 0 - 0 = 1 > 0.
  4. 4
    g(1)=cos⁡1−1≈0.540−1=−0.460<0g(1) = \cos 1 - 1 \approx 0.540 - 1 = -0.460 < 0.
  5. 5
    By IVT, there exists c∈(0,1)c \in (0,1) with g(c)=0g(c) = 0, i.e., cos⁡c=c\cos c = c.

Answer

By IVT, cos⁡x=x\cos x = x has a solution in (0,1)(0, 1).
Reformulate as 'a function equals zero' to set up IVT. Compute at endpoints, confirm a sign change, invoke continuity. The actual solution is near x≈0.739x \approx 0.739 (the Dottie number).

About Intermediate Value Theorem

If ff is continuous on the closed interval [a,b][a, b] and NN is any value between f(a)f(a) and f(b)f(b), then there exists at least one cc in (a,b)(a, b) such that f(c)=Nf(c) = N.

Learn more about Intermediate Value Theorem →

More Intermediate Value Theorem Examples

Example 1 easy

Show that [formula] has a root in the interval [formula].

Example 3 easy

Show that [formula] has a root in [formula].

Example 4 medium

Can IVT be applied to [formula] on [formula] to conclude it attains the value 0? Explain.

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