Lamps's question at Yahoo Answers about the Intermediate Value Theorem

[Fernando Revilla]

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]

Hello lamp,

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

$$f(-1)=\sin (-1)+6-5=1-\sin 1>0,\quad f(0)=-5<0$$

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

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

$$g(1)=-2<0,\quad g(2)=\ln 2+1>0$$

and we can reason as in (a).