082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

12.2 Maximum points and minimum points 407
y
A
y
A
(a)
B
Figure12.1
Thefunctionyhas a local maximum at Aand alocal minimumat B.
x
(b)
B
x
In Figure 12.1(a) tangents drawn at A and B would be parallel to thexaxis and so
atthesepoints dy
dx iszero.However,inFigure12.1(b)therearecornersatAandB.Itis
impossibletodraw tangents atthese points and so dy does notexistatthese points.
dx
Hence, when searching for maximum and minimum points we need only examine
thosepoints atwhich dy dy
is zero,or does not exist.
dx dx
Points at which dy is zero are known as turning points or stationary values of the
dx
function.
At maximumand minimumpoints either:
dy
(i) does notexist, or
dx
dy
(ii)
dx = 0
To distinguish between maximum and minimum points we can study the sign of dy
dx on
eithersideofthepoint.AtmaximumpointssuchasA,yisincreasingimmediatelytothe
leftofthepoint,anddecreasingimmediatelytotheright.Thatis, dy
dx ispositiveimmediatelytotheleft,and
dy
dx isnegativeimmediatelytotheright.Atminimumpointssuchas
B,yisdecreasingimmediatelytotheleftofthepoint,andincreasingimmediatelytothe
right.Thatis, dy
dy
isnegativeimmediatelytotheleft,and
dx dx ispositiveimmediatelytothe
right.Thisso-calledfirst-derivativetestenablesustodistinguishmaximafromminima.
This testcan be used even when the derivative does notexistatthe pointinquestion.
Thefirst-derivative testtodistinguish maximafromminima:
To the left ofamaximumpoint, dy dy
ispositive; tothe right,
dx dx isnegative.
To the left of a minimum point, dy
dy
isnegative; tothe right,
dx dx ispositive.