009A Sample Midterm 2, Problem 2

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:
What is a zero of the function $f(x)?$
A zero is a value $c$ such that $f(c)=0.$

Solution:

(a)

Step 1:
Intermediate Value Theorem
If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)}   is continuous on a closed interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,b]}
and $c$ is any number between Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(a)}   and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(b)} ,
then there is at least one number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} in the closed interval such that $f(x)=c.$

(b)

Step 1:
First, $f(x)$ is continuous on the interval $[0,1]$ since $f(x)$ is continuous everywhere.
Also,
$f(0)=2$
and         $f(1)=3-8+2=-3.$.

Step 2:
Since $0$ is between $f(0)=2$ and $f(1)=-3,$
the Intermediate Value Theorem tells us that there is at least one number $x$
such that $f(x)=0.$
This means that $f(x)$ has a zero in the interval $[0,1].$

Final Answer:
(a)     See solution above.
(b)     See solution above.

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