Question

Use the intermediate value theorem to show that the polynomial function has a zero in the given interval.
f(x)=9x^4-3x^2+5x-1;[0,1]

Answer 1

There are a number of ways to do this, lets go to the idiot's guide to Math...

f(0)=-1
f(1)=9-3+5-1=10

so how can one get to 10 from -1 by not crossing the y=0 axis?

Answer 2

Not sure that is why I asked :)

Answer 3

The IVT is dependent on the fact that f(x) is continuous. That is, f(x) <b>cannot</b> 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.

Answer 4

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 want-to-be types, so complex, it loses its meaning.