Question

use the intermediate theorem to show that the polynomial fuction f(x)=x^3+2x-1 has a zero on the interval [0,1]

Answer 1

The intermediate value theorem states that if some number N is between the values of f(a) and f(b) , then there exists some value c on the interval [a,b] such that f(c)=N

In this problem , your N is the zero of the function and c must be a number between 0 and 1.
\[f(0)=0+0-1=-1\]\[f(1)=1+2-1=2\]
0 is between the values of -1 and 2 so there indeed does exist some number on the interval [0,1] which yields f(c)=0