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Unformatted text preview: e must
increasing
be positi v e . We take the first derivative of f ’(x) which is f ”(x), the
positiv
a
second derivative of f(x), and compute f ”(a).
f ”(a ) > 0 implies a is minimum.
a
This is usually the easiest and most useful method. But there are exceptions.
117 1:
Exercise 1: For the function f(x) = x 3 f ’(x) = 3x 2 .
At x = 0 : f ’(0) = 0 implies zero is an extreme point.
Is x = 0 a maximum point or minimum point ?
Use all three methods to determine this.
Exercise 2: For the following functions find the extreme points and
determine if they are maximum or minimum.
a) x + 2
b) x 2
c)  x 2
d) x 3
e)  x 3
f)
x4
g)  x 4
From the number of times a function intersects the xaxis (the number of
real roots) can you estimate the number of extreme points ?
Exer
ercise
Ex er cise 3: Divide 10 into two par ts such that the product is a maximum.
Exer
ercise
Ex er cise 4: Divide 5 into two par ts such that the product is a maximum.
Draw the graph and verify.
Exer
ercise
Ex er cise 5: Divide 2 into two par ts such that the product is a maximum.
Draw the graph and verify. 118 Absolute Maximum and Absolute Minimum
It is possible to have more than one maximum and one minimum as in f(t) below.
maximum
minimum
maxim
minim
f(t) 0 t0 t1 t2 t3 t4 t5 t6 t7 t8 t In this case we will find the overall or absolute maximum and the overall or
absolute
absolute minimum . By solving f(t) = 0 we find all the maximum and minimum
maximum
minimum
points. Each maximum point is called a local maximum . And each minimum
maximum
maximum
minimum
point is called a local minimum .
minimum absolute maximum = maximum (all local maximum )
maximum
local maximum
maximu
absolute minimum = minimum (all local minimum )
minimum
local minimum
Exercise : Find the maximum and minimum points of f(t) = sin (+3ω0t) .
a
m
p
l
i
t
u
d
e f(t) = sin (+3ω0t)
T 119 At a maxim um minim um or inf lexion point, the tangent t o the curve is
maximum minimum
um,
infle
lexion
tangent
t ang
PARALLEL to the xaxis and hence tan θ = 0 . C onversely, if the tang ent t o the
tangent
t ang
cur ve at a point is...
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This note was uploaded on 11/29/2012 for the course PHYSICS 105 taught by Professor Tamerdoğan during the Fall '09 term at Middle East Technical University.
 Fall '09
 TAMERDOğAN

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