MATHEMATICS  THEORY



In engineering, there are many optimization problems
which can be reduced to finding the maximum or minimum values of a given function. In this section, the basic concept and theorem related to maximum and minimum are discussed. 





Maximum Value and Minimum Value

Example Description Diagram
Maximum and Minimum Value


Maximum
value is the absolute maximum value of the function in
its domain. In mathematics it is defined as following:
A function f has an absolute maximum at point x_{0} if
f(x_{0}) ≥ f(x)
for all x in its domain D. The number f(x_{0}) is called the maximum value of f on its domain. For example, the maximum value of the function plotted on the left is f(f) between a and h.
Similarly, minimum value is the absolute minimum value of the function in
its domain. In mathematics it is defined as following:
Function f has
an absolute minimum at point x_{0} if
f(x_{0}) ≤ f(x)
for all x in its domain D. The number f(x_{0}) is called the
minimum value of f on its domain. The minimum value for the function shown on the left is f(c) in [a,
h].
The maximum and minimum values of the function are called the extreme values of the function. In the function show on the left, f(c) and f(f) are the extreme values. 





Local Maximum and Local Minimum

Maximum Value
Minimum Value


The local maximum is defined in mathematics as:
A function f has a local maximum at x_{0} if
there is an open interval u that contains x_{0} such that f(x_{0}) ≥ f(x)
for all x in this interval u. Local maximum is also called relative
maximum. In Example Description Diagram, f(b), f(d) and f(f)
are the local maximum. Notice that f(f) is also absolute maximum and
extreme value of the function.
Similarly, local minimum is defined as,
Function f has local minimum at x_{0} if there
is an open interval u that contains x_{0} such that f(x_{0}) ≤ f(x)
for all x in this interval u. Local minimum is also called relative
minimum. In the previous example, f(c), f(e)
and f(g) are the local minimum. Notice that f(c) is also absolute minimum
and extreme value of the function.






Extreme Values

Function y = x^{2}
Function y = x^{3}


Some functions have extreme
value such as function y = x^{2}.
Since in (∞, +∞ ), f(x) ≥ f(0). Therefore, f(0) = 0 is
the absolute and local minimum value of this function. It is also the
extreme value. However, some functions do not have extreme value such
as function y = x^{3}. In the function's domain (∞,
+∞ ),
it does not have an absolute maximum or an absolute minimum value. In
reality, it has no local maximum and local minimum too. and thus function y =
x^{3} does not have extreme value. The Extreme Value Theorem has
been introduced in the
Continuity sectiion.
It gives the conditions under which a function has an extreme value.
Extreme Value 





Fermat's Theorem

Function y = (x^{2})^{0.5}


The Fermat's Theorem can be used to find the location
of an extreme value. The theorem states,
If a function has a local extreme which is the maximum or minimum at
x_{0}, and if df(x_{0})/dx exist, then df(x_{0})/dx
= 0.^{}
However, using Fermat's Theorem does not guarantee to find all extreme value. For example, although the absolute and local minimum value is 0 for function y = (x^{2})^{0.5}, but this value cannot be found by setting df(x_{0})/dx = 0. The reason is
df(0_{})/dx does not exist which has been shown in the
Differentiability section. Fermat's Theorem suggest that looking for extreme values at point x_{0} where
df(x_{0})/dx = 0 or df(x_{0})/dx
does not exist. Such points are given a special name  critical point. 





Critical Point



A critical point is the point where the derivative of the
function is 0 or does not exist. The method of calculating the critical
point can be understood by finding the critical point of function f(x)
= 6x^{1/3} 
5x.
Calculating the derivative of f(x) gives
df(x)/dx = d( 6x^{1/3})/dx  d(5x)
=
2x^{2/3 } 5
When df(x)/dx = 0
2x^{2/3 } 5 = 0
Therefore, x = 0.34
When df(x)/dx does not exist,
x^{2/3} = 0.
So x = 0
Thus, the critical point is at x = 0.34 and x = 0.
In terms of critical points, Fermat's Theorem can be rephrased as:
If a function has a local extreme at x_{0}, then x_{0} is
a critical point of the function. 





Method of Finding Absolute Maximum and Minimum



In order to find the absolute maximum and minimum of a continuous
function in its domain [a, b], the following steps need to be taken:
 Find the value of f(a) and f(b).
 Find the critical points of the function and calculate the value
at critical points.
 Compare the result of the above steps. The absolute maximum value
is the largest one and the absolute minimum is the smallest one.



