Question

How do you use the intermediate value theorem to verify that there is a zero in the interval [0,1] for `f(x)=x^3+3x-2`?

Answer 1

See the explanation below

Explanation:

The intermediate value theorem states that if `f(x)` is continuous on a closed interval `(a,b)`, and `c` is a number such that `f(a)<=c<=f(b)`, then there is a number `x` in the closed interval such that `f(x)=c`.
Here, `f(x)` is a polynomial function continous on the interval `(0,1)`
such that `f(0)=0+0-2=-2` and `f(1)=1+3-2=2`

Then there exists `0` such that `f(0) < 0 < f(1)`

There exists `x∈(0,1)` such that `f(x)=0`