Suppose f and g are differentiable functions such that x g(f(x)) f^ (g(x)) g^ (x)=f(g(x)) g^ (f(x)) f^ (x) for all real x . Also f is non negative and g is positive. If ₀^a f(g(x)) d x= 1 2 - e^ -2 a 2 for all reals a and g(f(0))=1 then the value of g(f(4)) is equal to e^ - where N , find the value of .

Step-by-step solution

aligned x f^ (g(x)) g^ (x) f(g(x)) & = g^ (f(x)) f^ (x) g(f(x)) x d d x ( f(g(x))) & = d d x ( g(f(x)))...(1) Also ₀^a f(g(x)) d x & = 1 2 - e^ -2 a 2 a aligned On differentiate, we get f(g(a))=e^ -2 a f(g(x))=e^ -2 x ...(2) From Eqns. (1) and (2), we get aligned x d d x ( e^ -2 x ) & = d d x ( g(f(x))) -2 x & = d d x ( g(f(x))) aligned On integrating, we get array ll & g(f(x))=-x^2+C & g(f(x))=k e^ -x^2 Given & g(f(0))=1 k=1 & g(f(4))=e⁻¹⁶ =16 array