I am a bit stuck on a chain rule question and would like to ask for some advice.
The question starts:
Q. The graph of y = (x^3 - x^2 +2)^3, is shown (point of inflection at x = -1, max at x=0 and min at unknown x=a point.
(i) Find the gradient.
OK so I use chain rule and get dy/dx = 3(x^3 - x^2 +2)^2 (3x^2 - 2x)
Then next part of question is:
(ii) Verify, showing your working clearly, that when x = -1, the curve has a point of inflection and when x = 0, the curve has a maximum.
I tackled this by saying stationary point when dy/dx = 0, then calculated that when x = -1, dy/dx does = 0. Same for x = 0.
But would I need to do 2nd order differential - ie d^2y/dx^2? Is that required. On above dy/dx is quite hard. Just want to know if there is an easier way.
Then next part asks:
(iii) the curve has a minimum when x = a. Find a and verify that this corresponds to a minimum.
solving this for dy/dx = 0 is quite tricky. How do I solve that? Or is there a way I am missing?