# How do you use the Intermediate Value Theorem to show that the polynomial function #f(x) = 10x^4 - 2x^2 + 7x - 1# has a root in the interval [-3, 0]?

See the explanation.

The Intermediate Value Theorem says that

Final Note In mathematics "there is a" mean "there is at least one". It does NOT mean "there is exactly one". .

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To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = 10x^4 - 2x^2 + 7x - 1 ) has a root in the interval [-3, 0], you need to show that the function changes sign over that interval.

First, evaluate ( f(-3) ) and ( f(0) ).

- ( f(-3) = 10(-3)^4 - 2(-3)^2 + 7(-3) - 1 )
- ( f(0) = 10(0)^4 - 2(0)^2 + 7(0) - 1 )

Then, determine the signs of ( f(-3) ) and ( f(0) ). If they are different, then by the Intermediate Value Theorem, there exists at least one root of the polynomial function in the interval [-3, 0].

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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