help with intermediate value theorem question

Question

Can someone please help me with this I dont know much about Intermediate Value Theorem could anyone please explain it to me so i can understand what exactly is being asked.

 

 

For f (x) = x3 – 3x2 – 2,

use the Intermediate Value Theorem to determine which interval must contain a zero of f.
A. Between 0 and 1
B. Between 1 and 2
C. Between 2 and 3
D. Between 3 and 4

Answer 1

Since f(x) is a polynomial function, it is continuous for all x.  

 

The Intermediate Value Theorem says that if f(x) is continuous on an interval [a,b] and if f(a)and f(b) have opposite signs, then f(x) = 0 for some number in the interval (a,b).  

 

Geometrically, if the graph of a continuous function lies below the x-axis at one endpoint of an interval and lies above the x-axis at the other endpoint of the interval, then the graph must cross the x-axis at least once between the interval's endpoints.  

 

For the given continuous function, f(3) = -2 < 0 and f(4) = 14 > 0.  So, by the Intermediate Value Theorem, f(c) = 0 for some number c in the interval (3,4).