For some function f(x) and g(x) which are differentiable x>0 satisfy the following condition. (i) ( f(x) x )^ =x^2 e^ -x^2 (ii) g(x)= 4 e^4 ₁^x e^ t^2 f(t) d t (iii) f(1)= 1 e Find the value of 3 e^4(f(2)-g(2)) .

Step-by-step solution

aligned g(x) & = 2 e^4 ₁^x 2 t e^ t^2 _ II f(t) _ I d t= 2 e^4 ( . f(t) t e^ t^2 |₁ ^x- ₁^x ( f(t) t ) e^ t^2 d t ) & = 2 e^4 ( f(x) x e^ x^2 -1- ₁^x t^2 d t )= 2 e^4 ( f(x) x e^ x^2 -1- 1 3 (x^3-1 ) ) aligned aligned g(x)= 2 e^4 ( f(2) 2 e^4-1- 8 3 + 1 3 )=f(2)- 20 3 e^4 (f(2)-g(2)) 3 e^4=20 aligned