Cambridge Additional Mathematics

-1

+ - +

3 x

f (x)'

Stationary point

where f^0 (a)=0

Sign diagram of f^0 (x)

near x=a

Shape of curve near x=a

local maximum

local minimum

stationary inflection

DEMO

or or

x=a

x=a

x=a x=a

y

x

stationary

inflection

local

maximum

local minimum

-2 1 3

+++-

local

maximum

local

minimum

stationary

inflection

1

-2-2 33

x

O

f (x)'

y = f(x)

a

+-

x

f (x)'

a

-+

x

f (x)'

a

--

x

f (x)'

a

++

x

f (x)'

376 Applications of differential calculus (Chapter 14)

SIGN DIAGRAMS

Asign diagramis used to display the intervals on which a function is positive and negative.

In calculus we commonly use sign diagrams of thederivative functionf^0 (x)so we can determine the nature

of a stationary point.

Consider the graph alongside.

The sign diagram of its gradient function is

shown directly beneath it.

We can use the sign diagram to describe the

stationary points of the function.

The signs on the sign diagram of f^0 (x)

indicate whether the gradient of y=f(x)

is positive or negative in that interval.

We observe the following properties:

Example 8 Self Tutor

Consider the function f(x)=x^3 ¡ 3 x^2 ¡ 9 x+5.

a Find they-intercept. b Find and classify all stationary points.

c Hence sketch the curve y=f(x).

a f(0) = 5, so they-intercept is 5.

b f(x)=x^3 ¡ 3 x^2 ¡ 9 x+5

) f^0 (x)=3x^2 ¡ 6 x¡ 9

=3(x^2 ¡ 2 x¡3)

=3(x¡3)(x+1) which has sign diagram:

cyan magenta yellow black

(^05255075950525507595)
100 100
(^05255075950525507595)
100 100 4037 Cambridge
Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_14\376CamAdd_14.cdr Monday, 7 April 2014 2:01:04 PM BRIAN