Cambridge Additional Mathematics

y

O x

f(x) = x 3

y

x

D,(6 18)

A,(-5 -16_ )Qw_

B,(-2 4)

C,(2 -4)

2

-2

y = f(x)

O

Applications of differential calculus (Chapter 14) 375

Astationary pointof a function is a point where f^0 (x)=0.

It could be a local maximum, local minimum, or stationary inflection.

TURNING POINTS (MAXIMA AND MINIMA)

Consider the following graph

which has a restricted domain

of ¡ 56 x 66.

Aisaglobal minimumas it has the minimum value ofyon the entire

domain.

Bisalocal maximumas it is a turning point where f^0 (x)=0and the

curve has shape.

Cisalocal minimumas it is a turning point where f^0 (x)=0and the

curve has shape.

Disaglobal maximumas it is the maximum value ofyon the entire

domain.

For many functions, a local maximum or minimum is also the global maximum or minimum.

For example, for y=x^2 the point (0,0) is a local minimum and is also the global minimum.

STATIONARY POINTS OF INFLECTION

It is not always true that whenever we find a value ofxwhere f^0 (x)=0, we have a local maximum or

minimum.

For example,

f(x)=x^3 has f^0 (x)=3x^2 ,

so f^0 (x)=0when x=0.

Thex-axis is a tangent to the curve which actually crosses over the curve

at O(0,0). This tangent is horizontal, but O(0,0) is neither a local

maximum nor a local minimum. It is called astationary inflectionas

the curve changes its curvature or shape.

B Stationary points

Points of inflection

are not required for

the syllabus.

Use of the words “local” and

“global” is not required for

the syllabus, but is useful for

understanding.

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_14\375CamAdd_14.cdr Tuesday, 8 April 2014 1:49:40 PM BRIAN