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If f(x)=|[(x-a)^4, (x-a)^3, 1] , [(x-b)^...

If `f(x)=|[(x-a)^4, (x-a)^3, 1] , [(x-b)^4, (x-b)^3,1] , [(x-c)^4, (x-c)^3,1]|` then `f'(x)=lambda|[(x-a)^4,(x-a)^3,1] , [(x-b)^4, (x-b)^3, 1] , [(x-c)^4, (x-c)^3,1]|`. Find the value of `lambda`

A

1

B

2

C

3

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
C
`f(x)=|((x-a)^(4),(x-a)^(3),1),((x-b)^(4),(x-b)^(3),1),((x-c)^(4),(x-c)^(3),1)|`
`f'(x)=|(4(x-a)^(3),(x-a)^(3),1),(4(x-b)^(3),(x-a)^(3),1),(4(x-c)^(3),(x-a)^(3),1)|+|((x-a)^(4),3(x-a)^(2),1),((x-b)^(4),3(x-b)^(2),1),((x-c)^(4),3(x-c)^(2),1)|+|((x-a)^(4),(x-a)^(3),0),((x-b)^(4),(x-b)^(3),0),((x-c)^(4),(x-c)^(3),0)|`
`=3|((x-a)^(4),(x-a)^(2),1),((x-b)^(4),(x-b)^(2),1),((x-c)^(4),(x-c)^(2),1)|`
`therefore" "lambda=3`
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