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# Local and Global(Absolute) Maxima and Minima

Local Maximum and local Minimum, Absolute maximum and absolute minimum

Local and Global(Absolute) Maxima and Minima

Definition 6.1

## Local Maximum and local Minimum

A function f has a local maximum (or relative maximum) at c if there is an open interval (a,b) containing c such that f(c)Ōēźf(x) for every x Ōłł (a,b)

Similarly, f has a local minimum at c if there is an open interval (a,b) containing c such that f(c) Ōēż f(x) for every x ! (a,b).

Definition 6.2

## Absolute maximum and absolute minimum

A function f has an absolute maximum at c if f(c)Ōēźf(x) for all x in domain of f. The number f(c) is called maximum value of f in the domain. Similarly f has an absolute minimum at c if f(c)Ōēżf(x) for all x in domain of f and the number f(c) is called the minimum value of f on the domain. The maximum and minimum value of f are called extreme values of f.

NOTE

Absolute maximum and absolute minimum values of a function f on an interval (a,b) are also called the global maximum and global minimum of f in (a,b).

## Criteria for local maxima and local minima

Let f be a differentiable function on an open interval (a,b) containing c and suppose that f ŌĆśŌĆÖ (c) exists.

(i) If f ŌĆś (c) = 0 and f ŌĆśŌĆÖ (c) > 0, then f has a local minimum at c.

(ii) If f ŌĆś (c) = 0 and f ŌĆśŌĆÖ (c) < 0,then f has a local maximum at c.

### NOTE

In Economics, if y = f(x) represent cost function or revenue function, then the point at which dy/dx = 0, the cost or revenue is maximum or minimum.

### Example 6.25

Find the extremum values of the function f(x)=2x3+3x2ŌĆō12x.

Solution :

Given

f(x) = 2x3+3x2ŌĆō12x ŌĆ” (1)

f ŌĆś (x) = 6x2+6xŌĆō12

f''(x) = 12x + 6

f ŌĆś (x) = 0 Ō¤╣ 6x2+6xŌĆō12 =0

Ō¤╣ 6(x2+xŌĆō2)= 0

Ō¤╣ 6(x+2)(xŌĆō1)= 0

Ō¤╣ x = ŌĆō2 ; x = 1

When

x= ŌĆō2

f ŌĆśŌĆÖ (ŌĆō2) = 12(ŌĆō2) + 6

= ŌĆō18 < 0

f(x) attains local maximum at x = ŌĆō 2 and local maximum value is obtained from (1) by substituting the value x = ŌĆō 2

f(ŌĆō2) = 2 (ŌĆō2)3+3(ŌĆō2)2ŌĆō12(ŌĆō2)

= ŌĆō16+12+24

= 20.

When

x = 1

f ŌĆśŌĆÖ (1) = 12(1) + 6

= 18.

f(x) attains local minimum at x = 1 and the local minimum value is obtained by substituting x = 1 in (1).

f(1) = 2(1) + 3(1) ŌĆō 12 (1)

= ŌĆō7

Extremum values are ŌĆō 7 and 20.

### Example 6.26

Find the absolute (global) maximum and absolute minimum of the function f(x)=3x5ŌĆō25x3+60x+1 in the interval [ŌĆō2,2]

### Solution :

f(x) = 3x5-25x3+60x+1 ŌĆ” (1)

fŌĆÖ (x) = 15x4ŌĆō75x2+60

= 15(x4ŌĆō5x2+4)

f ŌĆś (x) = 0 Ō¤╣ 15(x4ŌĆō5x2+4)= 0

Ō¤╣ (x2ŌĆō4)(x2ŌĆō1)= 0

x = ┬▒2 (or)  x = ┬▒1

of these four points ŌłÆ 2, ┬▒1 Ōłł [ŌłÆ2, 1]  and 2 Ōłē  [ŌłÆ2,1]

From (1)

f(ŌĆō2) =  3 ( - 2 )5 ŌĆō 25(- 2 )3 + 60 ( - 2 ) + 1

= ŌĆō15

When x= 1

f(1) =  3 ( 1 )5 ŌĆō 25(1 )3 + 60 ( 1 ) + 1

= 39

When x = ŌĆō 1

f(-1) =  3 ( -1 )5 ŌĆō 25(-1 )3 + 60 ( -1 ) + 1

= ŌĆō37.

Absolute maximum is 39

Absolute minimum is -37

Tags : Maxima and minima - Applications of Differentiation , 11th Business Mathematics and Statistics(EMS) : Chapter 6 : Applications of Differentiation
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11th Business Mathematics and Statistics(EMS) : Chapter 6 : Applications of Differentiation : Local and Global(Absolute) Maxima and Minima | Maxima and minima - Applications of Differentiation