Using differentiation, show that f(x) = 2x^3 - 12x^2 + 25x - 11 is an increasing function.

First compute the derivative of f(x) using the power rule on each term. f(x) = 2x^3 - 12x^2 + 25x - 11 so f'(x) = 6x^2 - 24x + 25. Now complete the square for the derivative. f'(x) = 6 * ((x-2)^2 - 4) + 25 = 6 * (x-2)^2 - 24 + 25 = 6 * (x-2)^2 + 1. Now observe that the first term is >= 0 since it is the result of a square multiplied by the positive constant 6. The second term, 1, is positive. Hence the whole expression is positive for any x. So we've shown that f'(x) > 0 for any x, so the function f(x) is increasing.