Can someone give me a hand? Here's what i have so far:

Question

Use the intermediate value theorem to determine if the function: f(x)=x^3+3x^2-9x-10
has a zero in the interval [-1, 0]

so this is what i have:
f(-1)=-1^3+3(-1^2)-9(-1)-10=-1-3-9-10= -23
and
f(0)=0^3+3(0^2)-9(0)-10= -10
What's next? I appreciate any assistance!

-Kriss

Answer 1

Ahem. Looks like it's time to brush up on your Algebra I
f(-1) = (-1)^3+3(-1)^2-9(-1)-10 = -1+3+9-10 = 1

Since f(0) changes sign on [-1,0] it has to assume the value 0 somewhere in that interval. So, yes it has a zero somewhere in there.

That is, since f is continuous, it cannot get from 1 to -10 without going through y=0 somewhere