IIT JEE , AIEEE Forum - Mathematics , Differential Calculus

sneha_91

if a₀+a₁/2+a₂/3........................a_n-1/n+a_n/n+1=0

show that the equ. a₀+a₁x+a₂x²........................a_nxⁿ=0has atleast one root in (0,1)

pottermania1990

for such qns...this is the standard approach,,,,,,,

xonsider the eqn. f(x)=.a0x + a1x^2+a2x^3....................

its simple application of mean value theorem...

f(0)=f(1)=0

since the function is continuous and differentaible in (0,1)

there lies atleat one c in (0,1) such that f'(c) = 0

proved

budokai_tenkaichi_returns

girl..

do u know lmvt..

its ok..then

integrate ur equation..

then u will see that the equation vanishes at x=0

and x=1 (from the condition given..)

this means F(0)=F(1)

hence ..f(x) which is the derivative of F(x) vanishes atleast once in (0,1) -----by LMVT

LMVT=lagrange's mean value theorem (cbse boards..)

RyuAmakusa

it is simple see take the integerated exp of the second eq. for this function f(0)=f(1) now apply rolles theorm.is it clear.

ashish_banga

let F(x) = a0 x + a0 x^2 /2 .........
F(0) = 0
F(1) = a0+a1/2+a2/3........................an-1/n+an/n+1=0

so here rolles theorum applies
there is a point in [0,1] where F'(c) = 0

so it has atleast a solution in [0,1]

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