Use the intermediate value theorem to determine whether the polynomial function has a zero in the given interval. f(x)= 3x^3 +10x +2; [-1,0]
Use the intermediate value theorem to determine if `f(x)=3x^3+10x+2` has a zero on [-1,0].
The intermediate value theorem states that if `f(x)` is a continuous function on a closed interval [a,b] and `f(a)<=k<=f(b)` or `f(a)>=k>=f(b)` then there exists some `c in (a,b)` such that `f(c)=k`
`f(x)` is a polynomial so it is everywhere continuous, so the intermediate value theorem applies. Thus on [-1,0] the function takes on all values from -9 to 2. Specifiacally, there is at least one `c in (-1,0)` such that `f(c)=0` , so there is at least one zero on the interval.