If f(x), g(x) be twice differentiable functions, satisfying f''(x) = g''(x), f'(1) = 2g'(1) = 4 and f(2) = 3g(2) = 9 then f(x) - g(x) at x = 4 is equal to

Step-by-step solution

Let h(x) = f(x) - g(x) . Given f''(x) = g''(x) , we have h''(x) = 0 . Integrating with respect to x , we get h'(x) = c₁ . We are given f'(1) = 4 and 2g'(1) = 4 g'(1) = 2 . Thus, h'(1) = f'(1) - g'(1) = 4 - 2 = 2 . Since h'(x) is constant, h'(x) = 2 . Integrating again with respect to x , we get h(x) = 2x + c₂ . We are given f(2) = 9 and 3g(2) = 9 g(2) = 3 . Thus, h(2) = f(2) - g(2) = 9 - 3 = 6 . Substituting x = 2 into h(x) = 2x + c₂ , we get 4 + c₂ = 6 c₂ = 2 . Therefore, h(x) = 2x + 2 . At x = 4 , h(4) = 2(4) + 2