Intermediate Value Theorem

Question

For which integer value of n does the Intermediate Value Theorem guarantee that f(x)=(x^3)+2x+4 has a zero in the interval [(n/2),((n+1)/2)]

Answer 1

We are looking for an integer n such that

f(n/2)<0

 

and

 

f((n+1)/2)>0.

 

The function f is always increasing, so we expect to have one zero.

 

Let n =0.

 

Then,

f(0) = 0³ + 2•0 + 4=4>0

 

This means n should be negative.

 

Let try n=-1.

f(-1/2)=(-1/2)³ + 2•(-1/2)+4

         =-1/8-1+4=2 7/8.

 

This means -1 could be our RIGHT endpoint if f(-1) is negative. If it is positive, we check the next value for n.

Can you finish the problem now?