IIT JEE , AIEEE Forum - Mathematics , Differential Calculus

ashgirl

i have a very big doubt........

can some 1 explain the terms in maxima and minima, increasing and decreasing function, how to find maxima, local maxima etc etc.....i get confused b/w these....

zane3004

srry if im wrong... maxima minima concepts are a little rusty...............

if a function is increasing, its slope of tangent or dx/dy is positive increases
if it is decreasing the slope of tangent dx/dy is negative and decreasing

to find maxima minima, think of a sine wave... the maxima will be at all pts with y=1
and minima y= -1
the slope of tangentthere will be 0 since it is horizontal.
:. dy/dx= 0 for any maxima/minima
now wen you have a function y=f(x) differentiate it and equate it to zero... you'll get some values of x which are the minima and maxima.
now double differentiate y=f(x) and substitute the values of x you got before... to see which is minima which is maxima.
if the value got on substituting is >0 it is minima and if it is <0 it is maxima.

local maxima, minima are for periodic functions like sine waves which have more than one maxima and minima. the lowest values are local maxima/minima

m.viddya

Working principle for finding maxima and minima

1.Let y=f(x) be a given function

calculate dy/dx=f ' (x) and equate it to zero...sove for x..say x=a,b

2.calculate d²y/dx²=f ' '(x)

3.for x=a...f ' ' (a)<0,then f(x)max at x=a and max value is f(a)

similarly for minimum...

mohit_jitani

this is not enough to say....dy/dx actually gives LOCAL maxima n minima of a function...u even need to check the end pts...all that we have said is applicable only to continuous funs....4 discntinuous funs u have to follow the basic definition of extremums.....one more thing always try to use 1st derivative test 4 chcking whether it is max or min...its more reliable n time saving

zane3004

and how do you find whether it is max. or min from 1st derivative ????

fwks_phoenix

find out the first derivative..

find d critical no.s..

for eg: if d critical no. is 1..

I. substitute x=0.9 (any value less than d critical no.) find out d sign of f(x)

II. x=1.1(any value greater than d critical no.) find out d sign of f(x)

if I is -ve and II is positive, it's minimum at x=1

if I is +ve and II is -ve,it's ,maximum at x=1...

fwks_phoenix

Second derivative test : just differentiate twice.. sub the critical no.s n see if's -ve or +ve...

-ve:maximum at dat pt.

+ve:minimum at dat pt.

apurviitjee2008

just like to add one point here
the above procedure is to find the local maxima or minima
to find the global maxima or minima
first u have to find the critical points of the function and the values of the function the the extreme values of domain
compare this with the values of the function given at other crtical points and u get the global maxima or global minima
note(the questions asked usually specify the domain of the function)

fwks_phoenix

yea i was telling abt local maxima/minima...

fwks_phoenix

off topic : how can we del a post that we make by mistake?

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