In this section we apply a theoretically important existence theorem called the Intermediate Value Theorem.

1 Intermediate Value Theorem

In this section, we introduce an important theorem related to continuous functions, the Intermediate Value Theorem. To build intuition, let us consider two real-life scenarios: the behavior of ambient temperature and the balance in a bank account. Suppose the temperature is at 8 a.m. and at noon. Because temperature changes continuously, we can be certain that at some time between 8 a.m. and noon, the temperature was exactly . This conclusion follows from the continuous nature of temperature: it does not jump abruptly from one value to another but passes smoothly through all intermediate values. Now, consider a different scenario involving a bank account. Suppose the account balance is $65 at 8 a.m. and $75 at noon. Can we similarly conclude that the account balance was exactly $70 at some time between 8 a.m. and noon? Unlike the temperature example, we cannot make this claim with certainty. Here’s why: It is possible that a single deposit of $10 was made at 11 a.m., causing the balance to jump directly from $65 to $75 without ever being exactly $70. On the other hand, if the $10 were added in smaller increments (e.g., two deposits of $5 each), the account could indeed have had a balance of $70 at some point. This uncertainty arises because bank account balances can change discontinuously due to individual transactions, which may create jumps in value. The fundamental difference between these two scenarios lies in the concept of continuity. Temperature changes continuously, while bank account balances can exhibit discontinuities. Continuity ensures that if a quantity changes from one value to another, it must pass through every intermediate value in between. In contrast, discontinuous changes allow for abrupt jumps, skipping over intermediate values entirely. Mathematically, this property is captured by the Intermediate Value Theorem (IVT):

The IVT formalizes our intuition about continuous change. It guarantees that for any continuous function, every intermediate value between and must occur at least once within the interval . In the examples above, the temperature behaves like a continuous function, making the IVT applicable and ensuring that the value was reached. In contrast, the bank account balance behaves like a discontinuous function, so the IVT cannot be applied, and we cannot draw the same conclusion.
The value in the theorem is called an intermediate value for the function on the interval .

The following figure illustrates the IVT.

(problem 1) Determine whether the IVT can be used to show that the equation has a solution in the open interval .

Is continuous on the closed interval ?

Yes No

and

Is an intermediate value?

Yes No

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

Yes No
(problem 2) Determine whether the IVT can be used to show that the equation has a solution in the open interval
Is continuous on the closed interval ?
Yes No

and

Is an intermediate value?

Yes No

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

Yes No
(problem 3a) Determine whether the IVT can be used to show that the function has a root in the open interval .
A root of a polynomial is a solution to the equation
Is continuous on the closed interval ?
Yes No

and

Is an intermediate value?

Yes No

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

Yes No
(problem 3b) Determine whether the IVT can be used to show that the function has a zero in the open interval .
A zero of a function is a solution to the equation
Is continuous on the closed interval ?
Yes No

and

Is an intermediate value?

Yes No

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

Yes No
(problem 4) Use the IVT to show that the equation has a solution in the open interval .
Rewrite the equation in the form
The functions and are continuous for all
Here is a detailed, lecture style video on the Intermediate Value Theorem:
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2025-01-19 16:31:36