Critical points

Critical Points

1. TARUN GEHLOT (B.E, CIVIL, HONOURS)
Critical Points
We will discuss the occurrence of local maxima and local minima of a function. In fact,
these points are crucial to many questions related to optimization problems. We will
discuss these problems in later pages.
Definition.A functionf(x) is said to have a local maximum atciff there exists an
intervalIaroundcsuch that
Analogously,f(x) is said to have a local minimum atciff there exists an
intervalIaroundcsuch that
A local extremum is a local maximum or a local minimum.
Using the definition of the derivative, we can easily show that:
Iff(x)has a local extremum atc, then either
These points are calledcritical points.

2. TARUN GEHLOT (B.E, CIVIL, HONOURS)
Example.Consider the functionf(x) =x
3
. Thenf'(0) = 0 but 0 is not a local extremum.
Indeed, ifx< 0, thenf(x) <f(0) and ifx> 0, thenf(x) >f(0).
Therefore the conditions
do not imply in general thatcis a local extremum. So a local extremum must occur at a
critical point, but the converse may not be true.Example.Let us find the critical points of
f(x) = |x
2
-x|
Answer.We have
Clearly we have
Clearly we have

3. TARUN GEHLOT (B.E, CIVIL, HONOURS)
Also one may easily show thatf'(0) andf'(1) do not exist. Therefore the critical points are
Letcbe a critical point forf(x). Assume that there exists an intervalIaroundc, that iscis
an interior point ofI, such thatf(x) is increasing to the left ofcand decreasing to the
right, thencis a local maximum. This implies that if for ( xclose
toc), and for ( xclose toc), thencis a local maximum. Note that
similarly if for ( xclose toc), and for ( xclose toc),
thencis a local minimum.
So we have the following result:
First Derivative Test. Ifcis a critical point forf(x), such thatf'(x) changes
its sign asxcrosses from the left to the right ofc, thencis a local extremum.
Example.Find the local extrema of
f(x) = |x
2
-x|
Answer.Since the local extrema are critical points, then from the above discussion, the
local extrema, if they exist, are among the points
Recall that
(1)

4. TARUN GEHLOT (B.E, CIVIL, HONOURS)
Forx= 1/2, we have
So the critical pointis a local maximum.
(2)
Forx= 0, we have
So the critical point 0 is a local minimum.
(3)
Forx= 1, we have
So the critical point -1 is a local minimum.

5. TARUN GEHLOT (B.E, CIVIL, HONOURS)
Letcbe a critical point forf(x) such thatf'(c) =0.
(i)
Iff''(c) > 0, thenf'(x) is increasing in an interval aroundc. Sincef'(c)
=0, then
f'(x) must be negative to the left ofc
and positive to the right
ofc. Therefore,cis a local minimum.
(ii)
Iff''(c) < 0, thenf'(x) is decreasing in an interval aroundc. Sincef'(c)
=0, then
f'(x) must be positive to the left ofc
and negative to the right
ofc. Therefore,cis a local maximum.
This test is known as the
Second-Derivative Test.
Example.Find the local extrema of
f(x) =x
5
- 5x.
Answer.First let us find the critical points. Sincef(x) is a polynomial function, thenf(x) is
continuous and differentiable everywhere. So the critical points are the roots of the
equationf'(x) = 0, that is 5x
4
- 5 = 0, or equivalentlyx
4
- 1 =0. Sincex
4
- 1 = (x-
1)(x+1)(x
2
+1), then the critical points are 1 and -1. Sincef''(x) = 20x
3
, then
The second-derivative test implies thatx=1 is a local minimum andx= -1 is a local
maximum.