###### Motivating Questions

How can continuity be used to show solutions to equations exist even when we cannot find the solutions?

How can continuity be used to show solutions to equations exist even when we cannot find the solutions?

In each part, sketch a graph of a function satisfying all the conditions or explain why it is implossible to satisfy all the conditions.

- The domain of \(f\) is the interval \([0,1]\text{,}\) \(f(0)=3\text{,}\) and \(f(1)=5\text{.}\)
- The domain of \(f\) is the interval \([0,1]\text{,}\) \(f(0)=3\text{,}\) \(f(1)=5\text{,}\) and \(f(x)\ne 4\) for all \(x\text{.}\)
- The domain of \(f\) is the interval \([0,1]\text{,}\) \(f(0)=3\text{,}\) \(f(1)=5\text{,}\) \(f(x)\ne 4\) for all \(x\text{,}\) and \(f\) is continuous.

There is a difference between finding something and showing that it must exist. Showing that something exists without finding it requires using conditions that you know and logical deduction. The following theorem is an example of showing something exists.

If

- \(f\) is continuous on a closed interval \([a,b]\) and
- \(f(a) \lt N \lt f(b)\) or \(f(a) \gt N \gt f(b)\text{,}\)

then there exists \(c\in (a,b)\) such that \(f(c)=N\text{.}\)

The proof of this theorem is too difficult for this course. Instead we will learn how to use this theorem in specific problems.

Consider the scenerio in Preview Activity 1.2.1 Item c. We are told that \(f\) is continuous on a closed interval \([0,1]\) and \(f(a)=1 \lt N=4 \lt f(b)=5\text{,}\) so the assumptions of the Intermediate Value Theorem 1.2.1 are satisfied. Therefore the conclusion of the theorem must hold, so there must be some \(x=c\) such that \(f(c)=4\text{.}\)

We want to show that the equation \(x^5+3=5x^3\) has a solution, but we cannot solve it by hand.

- Rearrange the equation to be of the form \(f(x)=0\text{.}\) Is \(f\) continuous?
- Is \(f(0)\) positive or negative?
- Find a value \(d\) so that \(f(d)\) has the opposite sign of \(f(0)\text{.}\)
- Show that the assumptions of the Intermediate Value Theorem 1.2.1 are satisfied. What is \(a\text{?}\) What is \(b\text{?}\) What is \(N\text{?}\)
- Apply the conclusion of the theorem. In what interval must \(x^5+3=5x^3\) have a solution?

Use the Intermediate Value Theorem to show that the equation \(x^2 =\cos(x)\) has a solution.

Determine whether each of the following are True or False. Explain your reasoning.

- If \(f(a) \lt N \lt f(b)\text{,}\) then there exists \(c\in (a,b)\) such that \(f(c)=N\text{.}\)
- If \(f\) is continuous on \([a,b]\text{,}\) \(f(a)=1\text{,}\) and \(f(b)=5\text{,}\) then there exists \(c\in (a,b)\) such that \(f(c)=0\text{.}\)
- If there exists \(c\in (a,b)\) such that \(f(c)=N\text{,}\) then \(f\) is continuous on the closed interval \([a,b]\) and either \(f(a) \lt N \lt f(b)\) or \(f(a) \gt N \gt f(b)\text{.}\)