The Extreme Value Theorem The Extreme Value Theorem

Introduction

The Extreme Value Theorem states that a function

f
, continuous on
[a,b]
, has a minimum and maximum value on that interval.

In mathematical notation, the Extreme Value Theorem is:
\text{Given } f \text{, continous on } [a,b] \\ \text{} \\ \exists c,d \in [a,b] \text{, where } f(c) \le f(x) \le f(d), \, \forall x \in [a,b]

Example

f(x)=x^2 \quad a=-1 \quad b=1
The red lines denote the interval.
f
has a minimum value at
(0,0)
and maximum values at
(-1,1)
and
(1,1)
.

Global Versus Local Extrema

A local minimum is a point where the function is less than its neighbors.
A local maximum is a point where the function is greater than its neighbors.

A global minimum is a point where the function is less than all other points on the function.
A global maximum is a point where the function is greater than all other points on the function.

f(x)=x^3-x

The grey dots are where
f
has a local minimum and a local maximum.
It has no global minimum or maximum.


f(x)=\sin(x)

The grey dots are where
f
has local minimums and local maximums.
These are also the global minimums and maximums.
f(x)=|x|

The grey dot is where
f
has both a local minimum and a global minimum.
There are no local or global maximums.

Critical Points

Local minimums and maximums can only exist where the derivative is either

0
or undefined. These are called critical points.

Local minimums and maximums can only at critical points because they are where the derivative can change its sign.
At a local minimum, the function is increasing before it and decreasing after. The derivative is positive before it and negative after.
At a local maximum, the function is decreasing before it and increasing after. The derivative is negative before it and positive after.

On the graphs, wherever there is a grey dot, the derivative is either at
0
or undefined.
Not all critical points are local minimums or maximums.

f(x)=x^3

f'(0)=0

x=0
is a critical point
There is no local minimum or maximum. The function is increasing before and after
x=0
.