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:

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)

Step 1:

Step 2:

Final Answer:
(a)
(b)

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