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Algebra

Blazing goIITian

 Joined: 20 Jul 2008 Post: 610
24 May 2009 11:37:01 IST
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PLZ try this .......................
Engineering Entrance , JEE Main , JEE Advanced , Mathematics , Algebra

The first term of an arithmetic progression is equal to unity . At what value of the common difference is

a1a3+a2a3    at a minimum . Give the full explanation.

Blazing goIITian

Joined: 7 May 2007
Posts: 1724
24 May 2009 11:54:04 IST
2 people liked this

let the common difference = d be x.

first term =

so,

so,

now,

f ' (x)

so,

now, f ' ' (x) = 4. it is positive. so, the value of x obtained is minima, and the value of f(x) should be minimum.

so, the common differene should be

Scorching goIITian

Joined: 25 Jan 2009
Posts: 246
24 May 2009 11:54:06 IST
1 people liked this

so a1= 1

a2= 1+d

a3= 1+2d

now y= a1a3+a3a2 = 1+2d+ 1+d+2d+2d2= 2d2+5d+2

now ,  differentiating 'y' w.r.t to x  and equating to 0(maxima and minima)

4d+5 = 0   thus d= -5/4

thusa1a3+a3a2 is minimum for d= -5/4

Scorching goIITian

Joined: 25 Jan 2009
Posts: 246
24 May 2009 11:55:10 IST
0 people liked this

hey this is wrong . i lost because of my slow typing.

Blazing goIITian

Joined: 7 May 2007
Posts: 1724
24 May 2009 11:55:25 IST
0 people liked this

let the common difference = d be x.

first term =

so,

so,

now,

f ' (x)

so,

now, f ' ' (x) = 4. it is positive. so, the value of x obtained is minima, and the value of f(x) should be minimum.

so, the common differene should be

Hot goIITian

Joined: 14 Jul 2008
Posts: 178
24 May 2009 12:09:05 IST
0 people liked this

given that,

a1=1

now let us assume that common difference is d

hence ,

a2=a1+(2-1)d = 1+d

a3=a1+(3-1)d=1+2d

a1a3+a2a3=2+5d+2d2

we hav to minimize it....

diff. it with rspct to d ,we get

4d+5

now critical point is

f'(d) =0

d=-5/4

using 2nd derivative test

f"(d)=4>0

hence it refers to point of minima.therefore d=-5/4

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