Consider the function f ( x )=( x -1)^2( x +1)( x -2)^3 . Let f(x) and g(x) be twice differentiable functions on [0,2] satisfying f ^ ( x )= g ^ ( x ), f ^ (1)=4, ~g ^ (1)=6, f(2)=3 and g(2)=9 . Then what is f(x)-g(x) at x=4 equal to ?

Step-by-step solution

f^ (x)=g^ (x) integrate b|s w.r.t. (x) f^ (x)+C₁=g^ (x)+C₂ ...(1) Putting x=1 aligned & f^ (1)+C₁=g^ (1)+C₂ & C₁-C₂=g^ (1)-f^ (1) & C₁-C₂=6-4=2 aligned again integrating equation (1) f(x)+C₁ x+a₁=g(x)+C₂ x+a₂ ...(3) Putting x=2 aligned & f(2)+C₁ 2+a₁=g(2)+2 C₂+a₂ & a₁-a₂=g(2)+2 C₂-2 C₁-f(2) & =9-3-2 (C₁-C₂ ) & =6-2 2=2 & a ₁- a ₂=2 aligned Rearranging equation (3) again, we get f(x)-g(x)=-x (C₁-C₂ )- (a₁-a₂ ) Putting x=4 , we get aligned [f(x)-g(x)]_ x=4 & =-4 (C₁-C₂ )- (a₁-a₂ ) & =-4 (2)-2 [f(x)-g(x)]_ x=4 & =-10