Use the intermediate value theorem to determine whether the polynomial function has a zero in the given interval. f(x)=4x^4+10x^3+4x+5; [-3,-2]

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Use the intermediate value theorem to determine whether the polynomial function `f(x)=4x^4+10x^3+4x+5` has a zero on [-3,-2].

The intermediate value theorem states that if a function is continuous on an interval [a,b], and f(a)<k<f(b) or f(a)>k>f(b), then there exists at least one c in [a,b] such that f(c)=k.

(1) Polynomials...

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Use the intermediate value theorem to determine whether the polynomial function `f(x)=4x^4+10x^3+4x+5` has a zero on [-3,-2].

The intermediate value theorem states that if a function is continuous on an interval [a,b], and f(a)<k<f(b) or f(a)>k>f(b), then there exists at least one c in [a,b] such that f(c)=k.

(1) Polynomials are continuous everywhere, so f(x) is continuous on [-3,-2].

(2) f(-3)=47

(3) f(-2)=-19

Then since 47>0>-19, there exists a c in [-3,-2] such that f(c)=0.

Therefore f(x) has at least one zero in the interval.

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