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)^2,1] , [(x-b)^4, (x-b)^2, 1] , [(x-c)^4, (x-c)^2,1]|`. Find the value of `lambda`

Options

A. 1
B. 2
C. 3
D. None of these

Answer

C

Explanation

`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`