Cambridge Additional Mathematics

We need to include points

where is undefined as critical

values of the sign diagram.

fx()

y

O x

5

(-1 10),

(3 -22),

y=x -3x -9x+5 32

-1

• 
  
  
  
  • 0 x
  
  
  
  


• 
  
  
  
  • f (x)'
    1
  
  
  
  

Applications of differential calculus (Chapter 14) 377

So, we have a local maximum at x=¡ 1 and a local minimum at x=3.

f(¡1) = (¡1)^3 ¡3(¡1)^2 ¡9(¡1) + 5 = 10

f(3) = 3^3 ¡ 3 £ 32 ¡ 9 £3+5=¡ 22

) there is a local maximum at (¡ 1 ,10) and a local minimum at (3,¡22).

c

Example 9 Self Tutor

Find and classify all stationary points of f(x)=

x^2 +1

x

.

=

x^2 +1

x

) f^0 (x)=

2 x(x)¡(x^2 +1)

x^2

=

x^2 ¡ 1

x^2

=

(x+ 1)(x¡1)

x

f^0 (x) has sign diagram:

So, we have local minima when x=§ 1.

f(¡1) =

(¡1)^2 +1

(¡1)

=¡ 2 and f(1) =

12 +1

1

=2

) there are local minima at (¡ 1 ,¡2) and (1,2).

SECOND DERIVATIVES AND STATIONARY POINTS

The second derivative of a function can be used to determine the nature of its stationary points.

For a function with a stationary point at x=a:

² If f^00 (a)> 0 , then it is alocal minimum.

² If f^00 (a)< 0 , then it is alocal maximum.

² If f^00 (a)=0, then it could be alocal maximum,alocal minimum,orastationary inflection

point.

f(x)

4037 Cambridge

cyan magenta yellow black Additional Mathematics

(^05255075950525507595)
100 100
(^05255075950525507595)
100 100
Y:\HAESE\CAM4037\CamAdd_14\377CamAdd_14.cdr Tuesday, 8 April 2014 1:52:53 PM BRIAN