Mathematics - Application of Derivative Question with Solution | TestHub

f(x) = x(x-1)(x-2)...(x-50) has 51 distinct real roots 0, 1, 2, ..., 50. By Rolle's theorem, f'(x) = 0 has exactly one root in each interval (k, k+1) for k = 0, 1, ..., 49, so there are exactly 50 critical points in (0, 50). On the interval (0,1) the factor x is positive and the other 50 factors are negative, so f(x) > 0 there, while f(0) = f(1) = 0; hence the first critical point is a local maximum. The sign of f alternates on successive intervals, so the critical points alternate maximum, minimum, maximum, ... Of the 50 intervals, f is positive on 25 and negative on 25, giving 25 local maxima and 25 local minima. Hence m = n = 25. Answer: option D.