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How do you use the Intermediate Value Theorem to show that the polynomial function # f(x) = x^2 − x + 1# has a root in the interval [-1, 6]?

Answer 1

The function can be written as

#f(x)=x^2-x+1=(x-1/2)^2+3/4>0# so it does not have any roots in #[-1,6]# because if she did we should have #f(x)=0# which is a contradiction.

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Answer 2

Samantha Adkison

To use the Intermediate Value Theorem to show that the polynomial function ( f(x) = x^2 - x + 1 ) has a root in the interval [-1, 6], you need to demonstrate that the function changes sign over that interval.

First, evaluate ( f(-1) ) and ( f(6) ). ( f(-1) = (-1)^2 - (-1) + 1 = 1 + 1 + 1 = 3 ) ( f(6) = (6)^2 - 6 + 1 = 36 - 6 + 1 = 31 )

Since ( f(-1) = 3 ) and ( f(6) = 31 ), and the function is continuous, by the Intermediate Value Theorem, there exists at least one value ( c ) in the interval [-1, 6] such that ( f(c) = 0 ), meaning the function has a root in that interval.

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Answer from HIX Tutor

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.