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Maxima and Minima

Maximum and Minimum values of a function in its domain

Maximum

Let f(x) be a function with domain \displaystyle D\subset R . Then f(x) is said to attain the maximum value at a point \displaystyle a\quad \epsilon \quad D  \displaystyle if\quad f\left( x \right) \quad \le \quad f\left( a \right)\quad for\quad all\quad x\quad \epsilon \quad D

In such a case, a is called the point of maximum and f(a) is known as the maximum value or the greatest value or the absolute maximum value value of f(x).

Consider the function

\displaystyle f\left( x \right) =-{ \left( x-1 \right) }^{ 2 }+10\quad for\quad all\quad x\quad \epsilon \quad R\\ \because -{ \left( x-1 \right) }^{ 2 }\quad \le \quad 0\quad for\quad all\quad x\quad \epsilon \quad R\\ \therefore -{ \left( x-1 \right) }^{ 2 }+10\quad \le \quad 10\quad for\quad all\quad x\quad \epsilon \quad R\\ \Rightarrow f\left( x \right) \quad \le \quad 10\quad for\quad all\quad x\quad \epsilon \quad R

Thus, 10 is he maximum value of f(x). Clearly f(x) attains this value at x=1. So x=1 is the point of maximum or the point of absolute maximum.

Minimum

Let f(x) be a function with domain \displaystyle D\subset R . Then f(x) is said to attain the maximum value at a point \displaystyle a\quad \epsilon \quad D  \displaystyle if\quad f\left( x \right) \quad \ge \quad f\left( a \right) \quad for\quad all\quad x\quad \epsilon \quad D

In such a case, a is called the point of minimum and f(a) is known as the minimum value or the least value or the absolute minimum value value of f(x).

Consider the function

\displaystyle f\left( x \right) ={ \left( 3x-1 \right) }^{ 2 }+5\quad for\quad all\quad x\quad \epsilon \quad R\\ \because { \left( 3x-1 \right) }^{ 2 }\quad \ge \quad 0\quad for\quad all\quad x\quad \epsilon \quad R\\ \therefore { \left( 3x-1 \right) }^{ 2 }+5\quad \ge \quad 5\quad for\quad all\quad x\quad \epsilon \quad R

Thus, 5 is he minimum value of f(x). Clearly f(x) attains this value at \displaystyle x=\frac { 1 }{ 3 }   . So \displaystyle x=\frac { 1 }{ 3 }   is the point of minimum or the point of absolute minimum .

Example

Find the maximum and minimum values if any, of the following functions

f(x)=sin 3 x +4

Solution: 

\displaystyle we\quad have\quad f\left( x \right) =\sin { 3x } +4\quad for\quad all\quad x\quad \epsilon \quad R\\ Now,\quad -1\quad \le \quad \sin { 3x } \quad \le \quad 1\quad for\quad all\quad x\quad \epsilon \quad R\\ \Rightarrow -1+4\quad \le \quad \sin { 3x } +4\quad \le \quad 1+4\quad for\quad all\quad x\quad \epsilon \quad R\\ \Rightarrow 3\quad \le \quad \sin { 3x } +4\quad \le \quad 5\quad for\quad all\quad x\quad \epsilon \quad R\\ \Rightarrow 3\quad \le \quad f\left( x \right) \quad \le \quad 5\quad for\quad all\quad x\quad \epsilon \quad R

Thus, the maximum value of f(x) is 5 and the minimum value is 3

\displaystyle Now,\quad f\left( x \right) =5\\ \Rightarrow \sin { 3x } +4\quad =5\\ \Rightarrow \sin { 3x } =1\\ \Rightarrow 3x=\frac { \pi }{ 2 } \\ \Rightarrow x=\frac { \pi }{ 6 }

\displaystyle And\quad f\left( x \right) =3\\ \Rightarrow \sin { 3x } +4\quad =3\\ \Rightarrow \sin { 3x } =-1\\ \Rightarrow 3x=-\frac { \pi }{ 2 } \\ \Rightarrow x=-\frac { \pi }{ 6 } 

 

Local Maxima and Minima

In the above section, we have talked about the greatest (maximum) and the least (minimum) values of a function in its domain. But there may be points in the domain of a function where the function does not attain the greatest (or the least) value but the values at these points are greater than or less than the values of the function at the neighbouring points. Such points are known as the points of local minimum or local maximum and we will be mainly discussing about the local maximutm and local minimum values of a function.

Local Maxima

A function f(x) is said to attain a local maximum at x = a if there exists a neighbourhood \displaystyle \left( a-\delta ;\quad a+\delta \right)  of a such that

\displaystyle f\left( x \right) \quad <\quad f\left( a \right) \quad for\quad all\quad x\quad \epsilon \quad \left( a-\delta ;\quad a+\delta \right)  x\neq a\\ f\left( x \right) -f\left( a \right) \quad <\quad 0\quad for\quad all\quad x\quad \epsilon \quad \left( a-\delta ;\quad a+\delta \right) x\neq a

In such a case f(a) is called the local maximum value of f(x) at x =a.

Local Minima

A function f(x) is said to attain a local minimum at x = a if there exists a neighbourhood \displaystyle \left( a-\delta ;\quad a+\delta \right)  of a such that

\displaystyle f\left( x \right) \quad >\quad f\left( a \right) \quad for\quad all\quad x\quad \epsilon \quad \left( a-\delta ;\quad a+\delta \right) \quad x\neq a\\ f\left( x \right) -f\left( a \right) \quad >\quad 0\quad for\quad all\quad x\quad \epsilon \quad \left( a-\delta ;\quad a+\delta \right)  x\neq a

The value of the function at x= a i.e., f(a) is called the local minimum value of f( x) at x = a.

The points at which a function attains either the local maximum values or local minimum values are known as the extreme points or turning points and both local maximum and local minimum values are called the extreme values of f(x) . Thus a function attains an extreme value at x=a if f(a) is either a local maximum value or a  local minimum value. Consequently at an extreme point ‘a’, f(x ) – f(a ) keeps the same sign for all values of x in a deleted neighbourhood of a. .
In Figure  we observe that the x-coordinates of the points A, C, E a re points
of local maximum and the values at these points i.e., their y-coordinates are the local maximum values of  f(x) . The x-coordinates of points B and D are points of local minimum and their y-coordinates are the local minimum values of f(x).

NOTE  By a local maximum (or local minimum) value of a function at a point x =a we mean the greatest (or the least) value in the neighbourhood of point x = a and not the absolute maximum (or the absolute minimum). In fact a function may have any number of points of local maximum ( or local  minimum) and even a local minimum value may be greater than a local maximum value. In Figure the minimum value at D is greater than the maximum value at A. Thus, a local maximum value may not be the greatest value and a local minimum value may bot be the least value of the function in its domain.

A NECESSARY CONDITION FOR EXTREME VALUES

We have the following theorem which we state without proof .

THEOREM A necessary condition for f(a) to be an extreme value of a function f(x) is that f ‘(a) = 0, in case it exists.

REMARK 1 This result states that if the derivative exists, it
must be zero at the extreme points.
A function may however attain an extreme value at a point without being derivable thereat. For example, the function f(x) = I x I attains the minimum value at the origin even x’ though it is not derivable at x = 0.

REMARK 2 This condition is only a necessary condition for the point x = a to be an extreme point. It is not sufficient i.e., f ‘(a)= 0 does not necessarily imply that x = a is all extreme point. There are functions for which the derivatives vanish at a point but do not have an extreme value thereat.

For example, for the function \displaystyle f\left( x \right) ={ x }^{ 3 },f`\left( 0 \right) =0  but at x = 0 the function does not attain an extreme value.

REMARK 3 Geometrically the above condition means that the tangent lo the curve y = f(x) at a point where the ordinate is maximum or minimum is parallel to the x-axis.

REMARK 4 As discussed in Remark 2 that all x, for which f'(x) = 0, do not give us the extreme values . The values of x for which f ‘(x) = 0 are called stationary values or critical values of x and the corresponding values of f(x) are called stationary or turning values of f(x).

As we have seen in Remark 2 that f'(a) = 0 is not the sufficient condition
for x = a to be an extreme point. The following  theorem provide  the sufficient condition for x = a to be an extreme point. This is known as the first derivative test and is stated without any proof .

First Derivative test for local maxima and minima

Let f(x) be a function differentiable at x = a. Then
(a) x = a is a point of local maximum of f (x) if
(i) f ‘(a) = 0 and
(ii) f ‘(x) changes sign from positive to negative as x passes through a i.e.,
f ‘(x) > 0 at every point in the left neighbourhood \displaystyle \left( a-\delta ,a \right)  of a and f`(x) > 0 at every point in the right neighbourhood \displaystyle \left( a,a+\delta \right)   of a

(b) x = a is a point of local minimum of f (x) if
(i) f ‘(a) = 0 and
(ii) f ‘(x) changes sign from positive to negative as x passes through a i.e.,
f ‘(x) < 0 at every point in the left neighbourhood \displaystyle \left( a-\delta ,a \right)  of a and f`(x) > 0 at every point in the right neighbourhood \displaystyle \left( a,a+\delta \right)   of a

(c) If f ‘(a)= 0, but f ‘(x) does not change sign, that is, f ‘(a) has the same sign in the complete neighbourhoodof a, then a is neither a point of local maximum nor a point of local minimum .

 

April 18, 2019
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