Not really sure how to go about doing this problem. For some reason the Intermediate value theorem has always given me issues and I got my final exam tomorrow. Thanks in advance!

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This post was edited by ChickenxChaser on Apr 29 2013 10:33pm

first off all look at the intermediate value theorem.
We need f to be continuous over the domain [a,b] and we are looking for a value between f(a) and f(b) lets call it u. we can then say their exists a number c which is an element of [a,b] and that f(c)=u

now we want to show that the root of our function is the cubed root of negative -2 . i.e we want to show that the cubed root of negative 2 exists.

now to find a root of a function we set the function =0
so f(x)=0
x^3=c
x=(c)^1/3
c could only be -2

so f(x)=x^3-(-2)

now we need to show that f(x)=0 lies between f(a) and f(b)

so lets just say b=3 a =-3
f(3)=27+2=29
f(-3)=-27+2=-25
so we can say that -25<0<29
i.e f(a)<0<f(b)
now by the intermediate value theorem there exists a c in [-3,3] s.t f(c)=0
with c equaling (-2)^(1/3)

hope that helps