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 :

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Correct answer: B. 40

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) = c_1. Given f'(1) = 4 and g'(1) = 2, h'(1) = f'(1) - g'(1) = 4 - 2 = 2. Therefore, c_1 = 2, which gives h'(x) = 2. Integrating again with respect to x, h(x) = 2x + c_2. 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) + c_2 = -6 c_2 = -10. Thus, h(x) = 2x - 10. Substituting x = 25, h(25) = f(25) - g(25) = 2(25) - 10 = 40. Answer: 40

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