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Let f(x) be a derivable function satisfying f(x+y)=f(x)+f(y) x, y R and f^ (0)=1 . If A= _ x 0 2^ f( x) -2^ f( x) x^3 , then find the value of e^ 4 A .

Correct answer

4

Step-by-step solution

Differentiating w.r.t. y keeping x constant, we get f^ (x+y) 1=f^ (y) Putting y=0, f^ (x)=1 f(x)=x+C Putting x=y=0 in given relation, f(0)=0 C=0 f(x)=x aligned A & = _ x 0 2^ x -2^ x x^3 = _ x 0 2^ x (2^ x- x -1 ) ( x- x) ( x- x x^3 ) & =1 2 _ x 0 x(1- x) x^3 = 2 2 aligned array lll & 4 A & =2 2= 4 & e^ 4 A & =4 array

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