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## AP®︎/College Calculus AB

### Course: AP®︎/College Calculus AB>Unit 1

Lesson 16: Working with the intermediate value theorem

# Intermediate value theorem review

Review the intermediate value theorem and use it to solve problems.

## What is the intermediate value theorem?

The intermediate value theorem describes a key property of continuous functions: for any function $f$ that's continuous over the interval $\left[a,b\right]$, the function will take any value between $f\left(a\right)$ and $f\left(b\right)$ over the interval.
More formally, it means that for any value $L$ between $f\left(a\right)$ and $f\left(b\right)$, there's a value $c$ in $\left[a,b\right]$ for which $f\left(c\right)=L$.
This theorem makes a lot of sense when considering the fact that the graphs of continuous functions are drawn without lifting the pencil. If we know the graph passes through $\left(a,f\left(a\right)\right)$ and $\left(b,f\left(b\right)\right)$...
... then it must pass through any $y$-value between $f\left(a\right)$ and $f\left(b\right)$.

## What problems can I solve with the intermediate value theorem?

Consider the continuous function $f$ with the following table of values. Let's find out where must there be a solution to the equation $f\left(x\right)=2$.
$x$$-2$$-1$$0$$1$
$f\left(x\right)$$4$$3$$-1$$1$
Note that $f\left(-1\right)=3$ and $f\left(0\right)=-1$. The function must take any value between $-1$ and $3$ over the interval $\left[-1,0\right]$.
$2$ is between $-1$ and $3$, so there must be a value $c$ in $\left[-1,0\right]$ for which $f\left(c\right)=2$.
Problem 1
$f$ is a continuous function.
$f\left(-2\right)=3$ and $f\left(1\right)=6$.
Which of the following is guaranteed by the Intermediate Value Theorem?

Want to try more problems like this? Check out this exercise.

## Want to join the conversation?

• If an equation of a cube root function is given and you are asked to find an interval that has least one solution, how would you go about that. I understand the Intermediate Value Theorem, but I'm not sure how to find the the interval given an equation.
• You find an x-coordinate where you know the function is negative and another where you know the function is positive. Then the interval with those endpoints must contain a solution, by Intermediate Value Theorem
• What if you're not given an interval to test within?

For example: Use the Intermediate Value Theorem to show that the equation x^3 + x + 1 = 0 has at least 1 solution.
• Then find an interval. By just picking x-values, we can see that your polynomial is positive at x=1 and negative at x= -1. So it must have a solution in (-1, 1). If you want a smaller interval than that, you can check the value at the halfway point. This will either force your zero into one half of (-1, 1) or the other, or it will actually find your zero. Once you have your new, smaller interval, you can continue to shrink it as much as you like.
• How does the intermediate value theorem work?
• The theorem basically says "If I pick an X value that is included on a continuous function, I will get a Y value, within a certain range, to go with it." We know this will work because a continuous function has a predictable Y value for every X value. By "predictable" I mean that the limits for a point from the right and left side are the same as the point's value in the function.

A simple way to imagine this is to pretend the continuous function occupies a box. We don't necessarily know what the function looks like, but we know where it can and can't go. Let's say our interval is [0, 10] and the end points are (0, 1) and (10, 5). In that case we know the function can't go any further left than 0, or any further right than 10. On the y-axis, our function could go lower than 1 or higher than 5 - we don't know. But the important thing is, it definitely has to go between one and five, otherwise it wouldn't connect our end points!

In answering a multiple-choice question about values given by the Intermediate Value Theorem, imagine this box on your graph. If the any option makes assumptions about what happens outside that box, don't select it, because there's no way for you to know those things from the information you've been given.
• what if you're asked to use IVT to show that x^1/2 + (x+1)^1/2 =4 has a root
• We have f(x) = x^(1/2) + (x+1)^(1/2) - 4

We need to show some closed interval [a, b] where f(x) is continuous.
We also need to show that 0 is between f(a) and f(b)

First lets establish a closed interval where the function is continuous.

f(x) is continuous for x >= 0 since the function is made by adding multiple square root functions which are also continuous for x>= 0.

Second, lets find a, and b by experimenting with different x-values.

f(0) = 0^(1/2) + (0+1)^(1/2) - 4
f(0) = 0 + 1 - 4
f(0) = -3

f(5) = 5^(1/2) + (5+1)^(1/2) - 4
f(5) is approximately 0.6856

Notice that 0 is between f(0) and f(5).

Finally, lets summarize the information we found.

We know that f(x) is continuous for x >= 0.
This means that f(x) is also continuous across [0, 5].
We found that f(0) = -3 and f(5) is approximately 0.6856
We now know that 0 is between f(0) and f(5).

Thus, f(x) has a root somewhere in the interval 0 <= x <= 5.

You could make a similar argument with other intervals that you find.
• One question asks "Why doesn't the IVT apply to g(x) on the interval [0, 4]. g(x)=2x^2-8x-5.

It's continuous, so why doesn't it apply?
• I think the point of the question is to notice that g(0)=g(4), so there are no values between them.
• what the IVT,IRCand ARC stands for
(1 vote)
• IVT= Intermediate Value Theorem
IRC= Instantaneous Rate of Change
ARC= Average Rate of Change
• How to show the existence of a real number whose square is two using intermediate value theorem?
(1 vote)
• The first question to ask is: What is the function? i.e. what is f(x)

The second question to ask is: What is a value for x that will make the function output a value less than 2? i.e. find a value for a so that f(a) < 2

The third question to ask is: What is a value for x that will make the function output a value greater than 2? i.e. find a value for b so that f(b) > 2

You then apply the intermediate value theorem to find that there must be a number between a and b that must give you 2 when squared.

Does that help?
• How would you solve this with a step by step answer:

Suppose that f is a continuous function on the interval [0,1] such that 0 smaller than or equal to f(x) is greater than or equal to 1 for each x in [0,1]. Show that there is a number c in [0,1] such f(c)=c
• I assume you mean 0 smaller than or equal to f(x) is smaller than or equal to 1 for each x in [0,1].

Define the function g(x) = f(x) - x. Because x is a continuous function, f(x) is a continuous function, and the difference of two continuous functions is continuous, g(x) is continuous.

Note that g(0) = f(0) and g(1) = f(1) - 1.

Since 0 <= f(x) <= 1 for each x in [0, 1], g(0) >= 0 and g(1) <= 0. Therefore, because g(x) is continuous, it follows from the intermediate value theorem that g(c) = 0 for some number c in [0,1]. Because g(c) = f(c) - c, f(c) = c for this number c.