Let f(x) and g(x) be twice differentiable functions satisfying f''(x) = g''(x) for all x R , f'(1) = 2g'(1) = 4 and g(2) = 3f(2) = 9 . Then f(25) - g(25) is equal to :

Step-by-step solution

Given f''(x) = g''(x) for all x R . Let h(x) = f(x) - g(x) . Taking the second derivative, h''(x) = f''(x) - g''(x) = 0 . Integrating with respect to x , h'(x) = c₁ . Given f'(1) = 4 and g'(1) = 2 , h'(1) = f'(1) - g'(1) = 4 - 2 = 2 . Therefore, c₁ = 2 , which gives h'(x) = 2 . Integrating again with respect to x , h(x) = 2x + c₂ . Given 3f(2) = 9 f(2) = 3 and g(2) = 9 , h(2) = f(2) - g(2) = 3 - 9 = -6 . Substituting x = 2 in h(x) , h(2) = 2(2) + c₂ = -6 c₂ = -10 . Thus, h(x) = 2x - 10 . Substituting x = 25 , h(25)