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be positi v e . We take the first derivative of f ’(x) which is f ”(x), the
second derivative of f(x), and compute f ”(a).
f ”(a ) > 0 implies a is minimum.
This is usually the easiest and most useful method. But there are exceptions.
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
g) -- x 4
From the number of times a function intersects the x-axis (the number of
real roots) can you estimate the number of extreme points ?
Ex er cise 3: Divide 10 into two par ts such that the product is a maximum.
Ex er cise 4: Divide 5 into two par ts such that the product is a maximum.
Draw the graph and verify.
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.
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 minimum . By solving f(t) = 0 we find all the maximum and minimum
points. Each maximum point is called a local maximum . And each minimum
point is called a local minimum .
minimum absolute maximum = maximum (all local maximum )
absolute minimum = minimum (all local minimum )
Exercise : Find the maximum and minimum points of f(t) = sin (+3ω0t) .
e f(t) = sin (+3ω0t)
T 119 At a maxim um minim um or inf lexion point, the tangent t o the curve is
PARALLEL to the x-axis and hence tan θ = 0 . C onversely, if the tang ent t o the
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