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lamps's question at Yahoo! Answers about the Intermediate Value Theorem

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661
Here is the question:

use the IVT to find the an interval of length one that contains a root of the equation
a) sin(x) = 6x + 5

b) ln(x) + x^2 = 3
Here is a link to the question:

Intermediate value theorem? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.
 

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661
Hello lamp,

(a) Denote [tex]f(x)=\sin x-6x-5[/tex]. Clearly, [tex]f[/tex] in continuos in [tex]\mathbb{R}[/tex]. We have:

[tex]f(-1)=\sin (-1)+6-5=1-\sin 1>0,\quad f(0)=-5<0[/tex]

Then, [tex]0\in (f(0),f(-1))[/tex] and according to the Intermediate Value Theorem there exists [tex]a\in (-1,0)[/tex] such that [tex]f(a)=0[/tex] or equivalently [tex]\sin a=6a+5[/tex]

(b) Now, denote [tex]g(x)=\ln x+x^2-3[/tex]. Clearly, [tex]f[/tex] in continuos in [tex](0,+\infty)[/tex]. We have:

[tex]g(1)=-2<0,\quad g(2)=\ln 2+1>0[/tex]

and we can reason as in (a).