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# The Intermediate Value Theorem

Functions that are continuous over intervals of the form , where and are real numbers, exhibit many useful properties. Throughout our study of calculus, we will encounter many powerful theorems concerning such functions. The first of these theorems is the Intermediate Value Theorem.

### The Intermediate Value Theorem

Let be continuous over a closed, bounded interval . If is any real number between and , then there is a number in satisfying . (See

(Figure)

).

### Application of the Intermediate Value Theorem

Show that has at least one zero.

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#### Solution

Since is continuous over , it is continuous over any closed interval of the form . If you can find an interval such that and have opposite signs, you can use the Intermediate Value Theorem to conclude there must be a real number in that satisfies . Note that

and

.

Using the Intermediate Value Theorem, we can see that there must be a real number in that satisfies . Therefore, has at least one zero.

### When Can You Apply the Intermediate Value Theorem?

If is continuous over , and can we use the Intermediate Value Theorem to conclude that has no zeros in the interval ? Explain.

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#### Solution

No. The Intermediate Value Theorem only allows us to conclude that we can find a value between and ; it doesn’t allow us to conclude that we can’t find other values. To see this more clearly, consider the function . It satisfies , and .

### When Can You Apply the Intermediate Value Theorem?

For and . Can we conclude that has a zero in the interval ?

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#### Solution

No. The function is not continuous over . The Intermediate Value Theorem does not apply here.

Show that has a zero over the interval .

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#### Solution

; is continuous over . It must have a zero on this interval.

#### Hint

Find and . Apply the Intermediate Value Theorem.

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