Let f(x) and g(x) be twice differentiable functions satisfying f''(x) - g''(x) = 2 for all x R . If the tangent to the curve y = f(x) - g(x) at x = 3 is given by the equation y = 4x - 5 , then the value of f(5) - g(5) is equal to :

Step-by-step solution

Let h(x) = f(x) - g(x) . Given that h''(x) = 2 . Integrating with respect to x , we get h'(x) = 2x + C . The tangent to the curve y = h(x) at x = 3 is y = 4x - 5 . The slope of this tangent is 4 , which means h'(3) = 4 . Substituting x = 3 into the derivative, h'(3) = 2(3) + C = 4 6 + C = 4 C = -2 . Thus, h'(x) = 2x - 2 . The point of tangency on the curve at x = 3 has the y -coordinate y = 4(3) - 5 = 7 . Therefore, h(3) = 7 . Integrating h'(x) with respect to x , we obtain h(x) = x^2 - 2x + D . Using h(3) = 7 , we