JEE Main201815 Apr 2018Morning ShiftMathematicsDeterminantsActual

If f(x)= | array ccc x & x & 1 2 x & x^2 & 2 x x & x & 1 array | , then _ x 0 f^ (x) x

Options

AExists and is equal to -2
BDoes not exist
CExist and is equal to 0
DExists and is equal to 2

Correct answer

A. Exists and is equal to -2

Step-by-step solution

aligned & f(x)= | array ccc x & x & 1 2 x & x^2 & 2 x x & x & 1 array | =& x (x^2-2 x^2 )-x(2 x-2 x x) =&-x^2 x-2 x x+2 x^2 x .+2 x x-x^2 x ) =& x^2 x-x^2 x & f^ (x)=2 x( x- x)+x^2 ( ^2 x+ x ) & _ x 0 f^ (x) x =& _ x 0 2 x( x- x)+x^2 ( ^2 x+ x ) x =& _ x 0 ( x- x)+x ( ^2 x+ x ) =& 2(0-1)+0=-2 So, _ x 0 f^ (x) x =-2 aligned

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