Page 21 - Math SL HB Sem 2

First Derivative Test for Local Extreme

Local

Maximum Minimum Inflexion
Sign of /'(x)when +0- -0+ +0+
Or
moving through a -0-
stationary point

Second Derivative Test for Local Extreme

Maximum Minimum Inflexion
Sign of /'(r)ar the <0 >0 0
stationary point (,Io)

Inflection points

A point wherc curvature changes from concave upward to concave downward (or
vice versa) is called point of inflection

/-\ For a curve (or part curve) which is concave downwards
in an interval S, /"(r) <0 for all x in S.
concave downward
ll For a curve (or part curve) which is concave upwards in
an interval S,f'(x) > 0 for all x in S.
concave upwards

lf f'(x) has a sign change on either sixe ofx : a, andf'(a) =.0, then
f
o We have a horizontal inflection if (a) : 0 also,
f
o We have a non-horimntal inflection if (a) + 0 also,

!| tangent .
sratronary inllection
Iangent slope = 0
/
," = /(.t) non-stationlrl inllcction l slopc + t)

If the tangent at a point of inflection lf the tangent at a point of inflection is
is horizontal, we say that we have a not horizontal we say that we have a
. horizontal or stationary inflection. non-horizcrtal or non-stationary
inllection.