How would you use the intermediate value theorem....?

Question

to determine if g(x)=4x^3-3x+3 has a zero between -2 and -1?

Answer 1

Notice that g(-2) = -23 and g(-1) = 2. The intermediate value theorem now says that, due to the continuity of the function g, there must be a point c between -2 and -1 such that g(c) = 0. I think of it like this: If I am on one side of the fence right here, and some distance down the road I am on the other side of the fence, then at some point between here and there I must have crossed over.

Answer 2

g is a polynomial so it's continuous on the whole R. g(-2) = -23 <0 and g(-1) = 2 >0. Since g is continuous on R, it's continuous on [-2 , -1]. Therefore, according to the intermediate value theorem, g takes on all the values between -23 and 2. This implies the existence of some some x* in (-2 , -1) such that g(x*) = 0.