Intermediate Value Theorem Problem

Question

The function f (x) = 3x7 −8x + 2 is a polynomial and therefore continuous everywhere.

(a) (12 points) State the Intermediate Value Theorem.

(b) (12 points) Use the Intermediate Value Theorem to show that f (x) has a zero in the interval [0, 1].

Answer 1

3x^7 -8x +2

f(0)=2

f(1) = 3-8+2 = -2

the sign change 2 to -2 means there's a zero between 0 and 1

x= 8/6 +/-(1/6)sqr(64-24)= 4/3 +/-(1/6)sqr40= 4/3+/-2sqr10 = about 2.4 or 0.3

0.3 is between 0 and 1

Intervediate Value Theorem aka Bolzano's Theorem says that if a confintuous function f(x) changes sign over an interval it has a zero in that interval