SOLUTION: Consider the function defined by f(x)=-2x^3-3x^2+5x-4. Use the intermediate value theorum to show that f has a zero between 0 and -1.

Sorry but f does not have a zero between 0 and -1.

Its graph looks like this and as you can see it doesn't cross 
the x-axis between 0 and -1, so it does not have a zero there.



You should point that out to your teacher. It must have been a typo.

However f does have a zero between -2 and -3 because 

INTERMEDIATE VALUE THEOREM
Let a and b be real numbers such that a < b. If f is a polynomial function such
that f(a) and f(b) are opposite in sign, then there exists at least one zero in
the interval (a, b).

[Notice that the other tutor, although he was lucky to have been correct in
saying f has no zero between 0 and -1, he did so by incorrectly assuming the
converse of the intermediate value theorem, which is not necessarily true.]  

Here a = -3 and b = -2.  f(-3) = 8 which is positive and f(-2) = -10, 
which is negative.  They are opposite in sign. Therefore, by the above theorem,
there exists at least one zero in the interval [a, b] which is the interval
(-3,-2).

The reasoning is that since the graph of a polynomial has no "loose ends", if
it is above the x axis at one place and below it at another place, it must
have crossed over the x-axis at least once between them.  It's like if you know
that somebody was in New York at noon and in New Jersey at 1 PM on the same
day, then they must have crossed the NY-NJ state line sometime between noon and
1 PM (since 1 hour is not enough time for them to have gone all the way around
to NJ through another state.) The other tutor's incorrect reasoning amounted to
"If a person is in NY at 12 noon and also in NY at 1 PM on the same day, then
they could not have crossed the NY-NJ state line during that hour", which is
false reasoning because they could have gone to NJ, turned around and come back
to NY, crossing the state line twice.)

Edwin