Chris claims that, “for any given value of x , the gradient of the curve y=2x^3 +6x^2 - 12x +3 is always greater than the gradient of the curve y=1+60x−6x^2” . Show that Chris is wrong by finding all the values of x for which his claim is not true.

(Question from AQA A level maths specimen papers) When looking at this question, you need to appreciate the steps required in order to reach the final answer before diving straight in. Firstly, the question clearly requires the gradients of both lines, so these will need to be found. Once this has been done, set up an inequality, showing the gradient of the first curve (y=2x^3 +6x^2 -12x +3) to be LESS than that of the second curve (the opposite of what Chris is claiming). The solution to this inequality will give the values of x that answer the question. dy1/dx = 6x^2+12x-12 dy2/dx= 60-12x The inequality is: 6x^2 + 12x - 12 < 60 - 12x Divide by 6 to simplify: x^2 + 2x - 2 < 10 - 2x x^2 + 4x -12 < 0 (x+6)(x-2)<0 Draw a diagram (will be done on whiteboard) to see clearly that -6 =< x =< 2 are the x-values for which Chris's claim are wrong.