Maxima & Minima, Slides of Mathematics

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INTRODUCTI Maxima & Minim

ON

In the previous chapter, we have used

derivatives to determine the intervals on which

function is increasing or decreasing. In this

chapter, we shall use derivatives to determine

points at which a function is maximum or

minimum and to find its maximum or minimum

values.

Maxima & Minim

INTRODUCTI

ON

In this chapter we discussed four terms :

LOCAL OR RELATIVE

MAXIMA

LOCAL OR RELATIVE

MINIMA

GLOBAL OR ABSOLUTE

MAXIMA

GLOBAL OR ABSOLUTE

MINIMA

LOCAL OR RELATIVE Maxima & Mini

MAXIMA

In mathematics, the largest value attained by the

function within a given neighborhood is called the local

(or relative) maximum value of a function.

A function is said to have

local maxima or relative

maxima at x = c , if is

maximum respectively in

comparison to its

neighbourhood.

f(c) >

f(c) >

Here x = c is point of local

Maxima & Minim

Local Maximum & Local

Minimum

In adjoin is the point of local

maxima. In adjoin is the point of local

minima. In adjoin is the point of local

maxima. In adjoin is the point of local

minima.

In adjoin is the point of local

maxima.

Maxima & Minim

Local Maximum & Local

Minimum

We also observed by adjoining

figureat the point , ,

at the point , ,

at the point ,

at the point ,

at the point , ,

Maxima & Minim

GLOBAL MAXIMA & GLOBAL

MINIMA

In mathematics, the largest

value attained by the function

in its entire domain is called

the absolute (or global )

maximum value of a function.

Similarly, the smallest value

attained by the function in its

entire domain is called the

absolute ( or global) minimum

𝐺𝑙𝑜𝑏𝑎𝑙 𝑀𝑎𝑥𝑖𝑚𝑢𝑚

𝐺𝑙𝑜𝑏𝑎𝑙 𝑀𝑖𝑛𝑖𝑚𝑢𝑚

𝐿𝑜𝑐𝑎𝑙 𝑀𝑎𝑥𝑖𝑚𝑖𝑚

𝐿𝑜𝑐𝑎𝑙 𝑀𝑖𝑛𝑖𝑚𝑢𝑚

Maxima & Minim

 (^) Condition is mathematically necessary

and sufficient conditions है जिजससे की

point x=a को point of maxima काहागे |

where x = a must be in domain of f(x).

पढ़ो ओर Maxima & Minim

 समझो Points of , not defined and end points of

the interval are called critical points.

Maxima/minima will occur at Critical

Points. यह कोई जरूरी नहीं है हिक सारे critical

points पर Maxima/ Minima define होता हो |

the end points of domain 𝒇 cannot be the

point of local maxima or local minima.

End point of interval be the point of

पढ़ो ओर Maxima & Minim

Points जिजन पर function हिक

value neighbourhood से बढ़ी

होती है , such points are

called Point of Local

Maxima or Relative

Maxima and Points जिजन पर

function हिक value

neighbourhood से छोटी होती है ,

such points are called

Point of Local Minima or

Relative Minima. A

function can more than

पढ़ो ओर Maxima & Minim

Continuous function मैं Maxima and Minima

alternatively occur होते है | हमेशा दो maxima के बीच एक

minima मिमलेगा ओर दो minima के बीच एक maxima मिमलेगा

Sometimes if question uses the term 'extremum' or

'extremal' or 'turning value' means point या तो

maximum का point है या minimum का point है |

पढ़ो ओर Maxima & Minim

यह जरूरी नहीं है हिक function की Local Maximum/ Minimum

value ही function की Greatest/Least value हो |

A function can have several maximum and minimum

values point minimum value point maximum value

greater From graph point x = d ZFT minima x = a

maxima greater@ I

पढ़ो ओर Maxima & Minim

Make sure, कभी हो सकता है हिक Greatest/Least

value interval के end point पर मिमले and end point

पर open interval हो then we say Greatest/Least

value of function is not achievable.

First Derivative Test For Local

Maxima & Minima

Derivative test at point x = a, means function x = a

पर Differentiable होना चहिहए |

From the graph above we observe that x = a is a

point of local maxima of function.