In this section we learn a theoretically important existence theorem called the Intermediate Value Theorem and we investigate some applications.
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 an important difference in their behavior. First, consider the ambient temperature and second, 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 sure that at some time between 8am and noon the temperature was exactly . Can we make a similar claim about money in a bank account? Suppose the account has $65 in it at 8am and then it has $75 in it at noon. Did it have exactly $70 in it at some time between 8 am and noon? We cannot answer that question with any certainty from the given information. On one hand, it is possible that a $10 deposit was made at 11am and so the total in the bank would have jumped from $65 dollars to $75 without ever being exactly $70. On the other hand, it is possible that the $10 was added in $5 increments. In this case, the account did have exactly $70 in it at some time. The fundamental reason why we can make certain conclusions in the first case but cannot in the second, is that temperature varies continuously, whereas money in a bank account does not (it will have jump discontinuities). When a quantity is known to vary continuously, then if the quantity is observed to have different values at different times then we can conclude that the quantity took on any given value between these two at some time between our two observations. Mathematically, this property is stated in the Intermediate Value Theorem.
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 the
interval . 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 a number between and such that .
The following figure illustrates the IVT.
First, the function is continuous on the interval since is a polynomial. Second, observe that and so that 10 is an intermediate value, i.e., Now we can apply the Intermediate Value Theorem to conclude that the equation has a least one solution between and . In this example, the number 10 is playing the role of in the statement of the theorem.
Is continuous on the closed interval ?
and
Is an intermendiate value?
Can we apply the IVT to conclude that the equation has a solution in the open interval ?
First, note that the function is continuous on the interval and hence it is continuous on the sub-interval, . Next, observe that and so that 2 is an intermediate value, i.e., Finally, by the Intermediate Value Theorem we can conclude that the equation has a solution on the open interval . In this example, the number 2 is playing the role of in the statement of the theorem.
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 the interval . 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 a root) on the open interval .
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 ?
Note that the equation is equivalent to the equation . The latter is the prefered form for using the IVT. So let Since is the difference between two continuous functions, it is continuous on the closed interval . Next, we compute and and show that 0 is an intermediate value: and and so, By the IVT, the equation has a solution in the open interval . Hence the equivalent equation has a solution on the same interval.