In this section we learn a theoretically important existence theorem called theIntermediate Value Theorem and we investigate some applications.

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Intermediate Value Theorem

In this section we discuss an important theorem related to continuous functions.Before we present the theorem, lets consider two real life situations and observe animportant difference in their behavior. First, consider the ambient temperature andsecond, consider the amount of money in a bank account.

First, suppose that the temperature is at 8am and then suppose it is at noon.Because of the continuous nature of temperature variation, we can be surethat at some time between 8am and noon the temperature was exactly .Can we make a similar claim about money in a bank account? Supposethe account has $65 in it at 8am and then it has $75 in it at noon. Did ithave exactly $70 in it at some time between 8 am and noon? We cannotanswer that question with any certainty from the given information. On onehand, it is possible that a $10 deposit was made at 11am and so the total inthe bank would have jumped from $65 dollars to $75 without ever beingexactly $70. On the other hand, it is possible that the $10 was added in $5increments. In this case, the account did have exactly $70 in it at some time. Thefundamental reason why we can make certain conclusions in the first casebut cannot in the second, is that temperature varies continuously, whereasmoney in a bank account does not (it will have jump discontinuities). When aquantity is known to vary continuously, then if the quantity is observed to havedifferent values at different times then we can conclude that the quantitytook on any given value between these two at some time between our twoobservations. Mathematically, this property is stated in the Intermediate ValueTheorem.

If the function is continuous on the closed interval and is a number between and ,then the equation has a solution in the open interval .

The value in the theorem is called an intermediate value for the function on theinterval . Note that if a function is not continuous on an interval, then the equation may or may not have a solution on the interval.

Remark: saying that has a solution in is equivalent to saying that there exists anumber between and such that .

The following figure illustrates the IVT.

Is continuous on the closed interval ?

and

Is an intermediate value?

Can we apply the IVT to conclude that the equation has a solution in the openinterval ?

First, note that the function is continuous on the interval and hence it iscontinuous on the sub-interval, . Next, observe that and so that 2 is anintermediate value, i.e., Finally, by the Intermediate Value Theorem wecan conclude that the equation has a solution on the open interval . Inthis example, the number 2 is playing the role of in the statement of thetheorem.

1.7 Intermediate Value Theoremap Calculus 2nd Edition

and

Is an intermendiate value?

Does the IVT imply that the equation has a solution in the open interval ?

Recall that a root occurs when . Since is a polynomial, it is continuous on theinterval . Plugging in the endpoints shows that 0 is an intermediate value: and so By the IVT, we can conclude that the equation has a solution (and hence has aroot) on the open interval .

Intermediate Value Theorem Examples

Is continuous on the closed interval ?

and

Is an intermendiate value?

Does the IVT imply that the function has a root in the open interval ?

All textbook readings are from:

Apostol, Tom M. Calculus, Volume 1: One-Variable Calculus, with An Introduction to Linear Algebra. Waltham, Mass: Blaisdell, 1967. ISBN: 9780471000051.

Additional course notes by James Raymond Munkres, Professor of Mathematics, Emeritus, are also provided.

Mean Value Theorem

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Extreme Value Theorem

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