Difference between revisions of "009A Sample Midterm 2, Problem 2"

Revision as of 12:37, 17 February 2017

The function $f(x)=3x^{7}-8x+2$ is a polynomial and therefore continuous everywhere.

a) State the Intermediate Value Theorem.

b) Use the Intermediate Value Theorem to show that

f(x)

has a zero in the interval

[0,1].

Foundations:

Solution:

(a)

Step 1:
Intermediate Value Theorem
If $f(x)$ is continuous on a closed interval $[a,b]$
and $c$ is any number between $f(a)$ and $f(b)$ ,
then there is at least one number $x$ in the closed interval such that $f(x)=c.$

(b)

Final Answer:
(a)
(b)

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+|'''Intermediate Value Theorem'''
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+|&nbsp; &nbsp; &nbsp; &nbsp; If <math style="vertical-align: -5px">f(x)</math>&thinsp; is continuous on a closed interval <math style="vertical-align: -5px">[a,b]</math>
-|}
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 2: &nbsp;
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+|&nbsp; &nbsp; &nbsp; &nbsp; and <math style="vertical-align: 0px">c</math> is any number between <math style="vertical-align: -5px">f(a)</math>&thinsp; and <math style="vertical-align: -5px">f(b)</math>,
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+&nbsp; &nbsp; &nbsp; &nbsp; then there is at least one number <math style="vertical-align: 0px">x</math> in the closed interval such that <math style="vertical-align: -5px">f(x)=c.</math>
 |}