the Intermediate Value Theorem guarantees that a zero exists between the two values.

Question

Evaluate f(x) = -x³ + 4x⁵ - 3x² + 1 at -1 and -3 to determine if the Intermediate Value Theorem guarantees that a zero exists between the two values.

a) f(-1) =    

b) f(-3) 

Answer 1

The intermediate value theorem states:

If is continuous on a closed interval [a,b] and c is any number between f(a) and f(b) inclusive, then there is at least one number x in the closed interval such that f(x) = c.

You function is:

f(x) = 4x⁵ -x³ - 3x² + 1

It is continuous on the interval [-3,-1]. The value of c we want is c = 0, that is f(x) = 0. Compute f(-3) and f(-1).

f(-3) = 4(-3)⁵ - (-3)³ - 3(-3)² + 1 = _____

f(-1) = 4(-1)⁵ - (-1)³ - 3(-1)² + 1 = _____

If one value is positive and the other negative then yes, it must pass though f(x)=0 somewhere in between. Otherwise, we don't know.