Use differentiation to show the function f(x)=2x^3–12x^2+25x–11 is an increasing function for all values of x

Firstly, I would remember what differentiation shows - which is the rate of change of a function. I would think about how I could use this to show it's always increasing. Now the first derivative would show it is increasing as if the gradient is always >0 it can be shown it is always increasing, doing what is required from the question.
So calculating the first derivative would give f'(x)=6x^2-24x+25. Now in this form it is not easy to tell if it is always increasing so we would need to manipulate it a bit to show what the question asks. You can factorise the first 2 terms into f'(x)=6(x^2-4x)+25. Next you can complete the square to give 6(x-2)^2-4+25=f'(x). Simplifying this gives f'(x)=6(x-2)^2+21. This is the final answer however you need to explain to the marker why this is correct and answers the question. You can explain the first part by saying no matter the value of x, when squared you always get a positive number. Meaning the first part is always positive. The second part can be shown by explaining that since it is positive, the value can only increase no matter if x=2 making it the smallest value possible (being 0) for the first part of the equation, and this can only increase by 21 answering the question showing it can only be increasing as required.