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]

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embizze | High School Teacher | (Level 1) Educator Emeritus

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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(-1)=(-1)^3+10(-1)+2=-9`

`f(0)=(0)^3+10(0)+2=2`

`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.

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