f(x) = 2x^3+6x^2-18x+1. For which values of x is f(x) an increasing function?

For f(x) to be an increasing function at x, the derivative of f(x) must be greater than 0. So, the answer to this question will be revealed by solving the equation f'(x) > 0 for x.By differentiating the terms of the polynomial f(x), we get f'(x) = 6x²+12x-18 > 0. To find the range(s) of x for which this is true, we can solve the quadratic 6x²+12x-18 = 0 to find the points at which it either starts or stops becoming true, and then use what we know about the graph to find the ranges.By first removing common factors from both sides:6(x²+2x-3) = 6(0)x²+2x-3 = 0and then factorising the quadratic:(x+3)(x-1) = 0we can get the solutions:x = -3, x = 1.So, by looking at the graph we can see that the regions we need are before the first intersection point (x = -3) and after the second (x = 1), so we get the solutions:x < -3, x > 1.