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Let "f(x)"|{:(pi^n,sinpix,cospix),((-1)^...

Let `"f(x)"|{:(pi^n,sinpix,cospix),((-1)^(n)!,-sin((npi)/2),-cos((npi)/2)),(-1,1/sqrt2,sqrt3/2):}|`
Then value or `d^n/(dx^n)["f(x)"]"at "x=1" is "`

A

0

B

1

C

-1

D

`1/sqrt2`

Text Solution

Verified by Experts

Know that,
`d^n/(dx^n)[pi^n/x]=((-1)^(n)!pi^n)/x^(n+1)`
`d^n/(dx^n)[sinpix]=pi^nsin(pix(npi)/2)" and "d^n/(dx^n)[cospix]=pi^nsin(pix+(npi)/2)`
Thus, `f^n(x)=|{:(((-1)^n n!pi^n)/(X^(n+1)),pi^nsin(pix+(npi)/(2)),pi^ncos(pix+(npi)/(2))),((-1)^n n!,-sin((npi)/(2)),-cos((npi)/(2))),(-1,(1)/(sqrt(2)),(sqrt(3))/(2)):}|`
`rArr" "f^n(x)=|{:((-1)^n n!pi^n,pi^nsin(pix+(npi)/(2)),pi^ncos(pix+(npi)/(2))),((-1)^n n!,-sin((npi)/(2)),-cos((npi)/(2))),(-1,(1)/(sqrt(2)),(sqrt(3))/(2)):}|`
`=0{ :.R_1" and " R_2 " are propartional"}`
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