Intermediate Value Theorem

Question

For f (x) = x^4 – 3x^2 − 8, use the Intermediate Value Theorem to determine which interval must contain a zero of f. Work/Explanation required

4. _______

A. Between 0 and 1

B. Between 1 and 2

C. Between 2 and 3

D. Between 3 and 4

Answer 1

The Intermediate Value Theorem states that on a closed interval [a,b], if there is a value, d, between f(a) and f(b), then there is a point, c, on the interval [a,b] such that f(c) = d.

 

So how does this relate to your problem?  Simple.  In your problem we want to know if there is a value f(x=c) = 0 on the interval.  So compute f(a) and f(b) for each of the candidate intervals and see if f(x)=0 lies between them.  If it does, it's the answer.

 

I'll do A (Between 0 and 1) for you:

 

f(0) = (0)⁴-3(0)²-8=-8

f(1) = (1)⁴-3(1)²-8 = 1-3-8 = -10

 

f(x) = 0 does not lie between f(0)=-8 and f(1)=-10, so choice A is not the answer.