Unformatted text preview: :
f (a  δ x) < f(a) and f (a+ δ x) < f(a)
a
a
a
a
implying that a is a maximum point
or
f (a  δx) > f(a) and f (a+δx) > f(a)
a
a
a
a
implying that a is a minimum point.
These calculations may be tedious.
If we know the graph of the function then we can determine whether a is
minimum.
maximum or minimum This Geometric m ethod may not be practical.
So let us try to find an Analytical method.
Analytical
Anal
We shall use the results :
if f(x) is increasing then f ’(x) is positive
positive.
if f ’(x) is positive then f(x) is increasing
increasing.
if f(x) is decreasing then f ’(x) is negative
negative.
if f ’(x) is negative then f(x) is decreasing .
decreasing.
116 Method 2 : At the maximum point a : f(x) is increasing before and
after.
decr easing after.
Correspondingly, f ’(x) is positi v e befor e and ne g a ti v e after.
positiv befor
ore
neg tiv after.
We need to check around a if :
f ’(a  δx) > 0 and f ’(a + δ x) < 0.
a
a
At the minimum point a : f(x) is decreasing before and
after.
incr easing after.
Correspondingly, f ’(x) is ne g a ti v e befor e and positi v e after.
neg tiv befor
ore
positiv after.
We need to check around a if :
f ’(a  δx) < 0 and f ’(a + δ x) > 0.
a
a
This again may be tedious.
maximum
positiv befor
ore
Method 3: We know that at the maxim um point a : f ’(x) is positi v e befor e
and ne g a ti v e after.
neg tiv after.
Going from positive to negative means f ’(x) must be decreasing.
We also know that if a function is decr easing its first derivative
decreasing
must be ne g a ti v e . We take the first derivative of f ’(x) which is f ”(x),
neg tiv
the second derivative of f(x), and compute f ”(a).
a
f ”(a) < 0 implies a is maximum.
a
Similarly, we know that at the minimum point a : f ’(x) is negative
befor
ore
positiv after.
bef or e and positi v e after.
Going from negative to positive means f ’(x) must be increasing.
We also know that if a function is incr easing its first derivativ...
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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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