Use the intermediate value
posted by Jenn on .
Use the intermediate value theorem to show that the polynomial function has a zero in the given interval.
f(x)=9x^43x^2+5x1;[0,1]

There are a number of ways to do this, lets go to the idiot's guide to Math...
f(0)=1
f(1)=93+51=10
so how can one get to 10 from 1 by not crossing the y=0 axis? 
Not sure that is why I asked :)

The IVT is dependent on the fact that f(x) is continuous. That is, f(x) cannot get from 1 to 10 without being 0 somewhere on the way.
If f is not continuous, then there might be a hole at f=0, so there would be no guarantee that f(c)=0 for some 0<c<1. 
Reread the intermediate value theorem, it concludes that one can't get to 10 from 1 with a continuous function without passing the y=0 axis. Often, the mean value, and intermediate value theorem are written in math texts by lawyer wanttobe types, so complex, it loses its meaning.