If f(x)= (x^2+e^x ) (x^4+e^ 2 x ) and _ x f(x)= and _ x - f(x)=m then find the value of [ 2 m ] is (Where [.] denote the greatest integer function)

Step-by-step solution

aligned & _ x (x^2+e^x ) (x^4+e^ 2 x ) = _ x e^x (1+ x^2 e^x ) e^ 2 x (1+ x^4 e^ 2 x ) & _ x x+ (1+ x^2 e^x ) 2 x+ (1+ x^4 e^ 2 x ) = 1+ 1 x (1+ x^2 e^x ) 2+ 1 x (1+ x^4 e^ 2 x ) as _ x x^2 e^x 0 as _ x x^2 e^ 2 x 0 & = 1 2 m= 1 2 [ 2 m ]= [ 2 1 2 1 2 ]=2 aligned