/
,festinq
fqr
local
frct+five)
Me
6)-petn*iq"'
f'will
have
a
local
maximum
at:*#ifanA ody
if t(x)<tia)
for
x's
near a.
\7-_---
'.-.-
-Hnition:
f will
have a local minimum
at:<=a if
and
only if t{x)>f(a}
for
x's
near
a.
-For
smooth
surves, these ocqrr
where
f
'(xI=O.
They are
called
critical
poiuts.
-Be
careful,
the converse
is
not necessarily
true
(covered
earlier).
The
First
Deriqativs
Tmt
faf,uax/qin
{usuafiy
used
in
very
simple $ituations)
-Key:
The
derivative
nust hsve
a changc in sign
arouad
&
max or
minpoint.
-If
f
'(x)>0
for
x<a
and
f
'(x)<0 for x>a
-+
N{aximum
point
-It
f
'{x)<0
fbr x<a
and
f
'(x)>0
for x>a
-+ Minimum
point
Shovr with
r'-:o'at:r0.
The 2d
llerivative
Te$t
fo{ mqs/miq
(more popular)(used
the most)(good
for
complicated eurves)
-lf
f
"(x)>0
whsre f
'(x)=0
-+
Minimum
poirt
Why?
*If
f
"(x)<0
where
f
'(x)-0
*+
Ma.ximum
point.
ldrhy?
Example:
Find locat
Max
ard lv{in
points
for
f(x}
:
x
t
-9x
'
-48x+52
&
justify
using
the 2d derivative
test.
?4"e.
tr*"
")**rffi*,*
"?tM.
'
fu,.'
'*r*
##)
fra:
s'&): ?
*-
tgv-
*?
*'t/x)
=
lr?<*
E
f
'tx\
=a
)
g
y!a&**4f
=n
y\ay*lg=A
t
?c+r)
{*-t)=o
Eie.o"t,
Bi
f
{/-z)=
to+,
{{r}
* '3f
A
L
94**e-
-*ffi*
)
J'?*)
=
d
+
Alt
*/F
--
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Xz3,{t't)=t44
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