Calculus and Vectors – How to get an A+

4.2 Critical Points. Local Maxima and Minima ©2010 Iulia & Teodoru Gugoiu - Page 1 of 2

4.2 Critical Points. Local Maxima and Minima A Local Maximum A function f has a local (relative) maximum at cx = if

)()( cfxf ≤ when x is sufficiently close to c (from both sides).

)(cf is called the local (relative) maximum value. ))(,( cfc is called the local (relative) maximum point.

B Local Minimum A function f has a local (relative) minimum at cx = if

)()( cfxf ≥ when x is sufficiently close to c (from both sides).

)(cf is called the local (relative) minimum value. ))(,( cfc is called the local (relative) minimum point.

C Note The following points are neither local minimum or maximum points.

Ex 1. A function is defined by the following graph. Find the local minimum or maximum points.

Local minimum points are )2,5(−A , )1,1(−B , and )2,5(C . Local maximum points are )4,2(−D and )5,2(E .

D Critical Numbers and Critical Points The number fDc∈ is a critical number if either

0)(' =cf or DNEcf )(' The point ))(,( cfcP is called critical point.

Ex 2. Find the critical points for: a) 23 32)( xxxf +=

132)1(3)1(2)1(0)0(

100)(')1(666)('

23

2

=+−=−+−=−=

−==⇒=+=+=

ff

xorxxfxxxxxf

The critical points are )0,0(A and )1,1(−B .

b) 3 2 1)( −= xxf

[ ]

0,101)('

1,00)(')1(3

2)2()1(31)1()('

2

3 223/22'3/12

=±=⇒=−⇒

−==⇒=

−=−=−= −

yxxDNExf

yxxfx

xxxxxf

The critical points are )1,0( −A , )0,1(−B , and )0,1(C .

Calculus and Vectors – How to get an A+

4.2 Critical Points. Local Maxima and Minima ©2010 Iulia & Teodoru Gugoiu - Page 2 of 2

E Fermat’s Theorem If f has a local extremum (minimum or maximum) at

cx = then c is a critical number for f . So, at a local extremum:

0)(' =cf or DNEcf )('

Note. If

0)(' =cf or DNEcf )(' then the function f may have or not a local extremum (minimum or maximum) at cx = .

F First Derivative Test Let c be a critical number of a continuous function f . If )(' xf changes sign from positive to negative at c then f has a local maximum at c .

If )(' xf changes sign from negative to positive at c then f has a local minimum at c .

If )(' xf does not change sign at c then f has no local extremum (minimum or maximum) at c .

Ex 3. Find the local extrema points for the functions in the previous example. a) 23 32)( xxxf +=

)1(666)(' 2 +=+= xxxxxf Sign Chart:

x 1− 0 )(xf _ 1 ` 0 _ )(' xf + 0 - 0 +

The point )0,0(A is a local minimum point and the point

)1,1(−B is a local maximum point.

b) 3 2 1)( −= xxf

3 22 )1(3

2)('−

=x

xxf

Sign Chart:

x 1− 0 1 )(xf ` 0 ` 1− _ 0 _ )(' xf - DNE - 0 + DNE +

The point )1,0( −A is a local minimum point. Ex 4. Find a function of the form dcxbxaxxf +++= 24)( with a local maximum at )6,0( − and a local minimum at

)8,1( − .

642)(

42022

0024860

)4(),3(),2(),1(

)4(0240)1(')3(00)0('

24)('

)2(88)1()1(66)0(

24

3

−−=∴

−==

⇒⎩⎨⎧

=+−=+

⇒⎩⎨⎧

=++−=−++

⇒

=++⇒==⇒=

++=

−=+++⇒−=−=⇒−=

xxxf

bandababa

babacbaf

cfcbxaxxf

dcbafdf

Reading: Nelson Textbook, Pages 172-178 Homework: Nelson Textbook: Page 178 #3ab, 5ad, 7adf, 10, 13, 15