The Intermediate Value Theorem

Warm-up

Three quick questions from earlier lessons.

Discover

At 7 a.m. the temperature was 4 °C; at noon it was 15 °C. Temperature changes continuously. Was it exactly 10 °C at some moment in between?

Yes, at some moment
Not necessarily

Key idea

A continuous function cannot skip values.

Intermediate Value Theorem (IVT): if

f

is continuous on

[a, b]

and

N

is between

f(a)

and

f(b)

, then

f(c) = N

for some

c

in

(a, b)

. In particular, if

f(a)

and

f(b)

have opposite signs,

f

has a zero in

(a, b)

.

Worked example

Show that $x^3 - 2x - 1 = 0$ has a solution between 1 and 2.

f(x) = x^3 - 2x - 1

is continuous on

[1, 2]

A polynomial is continuous everywhere.

f(1) = -2 < 0

and

f(2) = 3 > 0

Evaluate at the ends.

So

f(c) = 0

for some

c

in

(1, 2)

The IVT with N = 0.

Your turn, guided

Your turn: $f(x) = x^2 - 5$ on $[2, 3]$ .

f(2) = -1

4 − 5.

f(3) = 4

: a sign change, so a zero (

\sqrt{5}

) lies in between

9 − 5.

Practice ★

$f$ is continuous on $[1, 9]$ , with $f(1) = -1$ and $f(9) = 7$ . Must $f$ have a zero between 1 and 9?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing
$f$ is continuous on $[3, 5]$ , with $f(3) = -2$ and $f(5) = 2$ . Must $f$ have a zero between 3 and 5?
1. Yes, by the Intermediate Value Theorem
2. Only if $f$ is a polynomial
3. Only if $f$ is increasing

Watch out!

Practice ★★

$f(x) = \dfrac{1}{x}$ has $f(-4) < 0$ and $f(4) > 0$ . Does $f$ have a zero in $(-4, 4)$ ?
1. No: $f$ is not continuous on $[-4, 4]$ , so the theorem does not apply
2. Yes, by the Intermediate Value Theorem
3. Yes, at $x = 0$

$f(x) = x^3 + x - 3$ is continuous. Between which consecutive whole numbers $m$ and $m + 1$ does it have a zero? Give $m$ .
$f(x) = x^3 + x - 11$ is continuous. Between which consecutive whole numbers $m$ and $m + 1$ does it have a zero? Give $m$ .
$f(x) = x^3 + x - 31$ is continuous. Between which consecutive whole numbers $m$ and $m + 1$ does it have a zero? Give $m$ .

Exit ticket

Three questions to check you have got it.

$f(x) = x^3 + x - 3$ is continuous. Between which consecutive whole numbers $m$ and $m + 1$ does it have a zero? Give $m$ .
$f(x) = x^3 + x - 11$ is continuous. Between which consecutive whole numbers $m$ and $m + 1$ does it have a zero? Give $m$ .
$f(x) = x^3 + x - 31$ is continuous. Between which consecutive whole numbers $m$ and $m + 1$ does it have a zero? Give $m$ .

Summary card

The Intermediate Value Theorem

Check that $f$ is continuous on $[a, b]$ .
Compare $f(a)$ and $f(b)$ with $N$ (often 0).
If $N$ is between them, $f(c) = N$ for some $c$ in $(a, b)$ .

Watch out: no continuity, no conclusion.

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