Locate the position and the nature of any turning points in the function: 2x^3 - 9x^2 +12x
I would first ask the student to draw a cubic graph(preferably the function in asking depending on the student's capability). I would then ask the student to show me where the stationary points in the graphs are. All this will aid in the student's understanding of the underlying concepts of the first and second derivatives. If the student struggles at any point, I will give the student an opportunity to think before giving hints.
The first derivative is worked out: dy/dx= 6x2 -18x+12. This is set to equal zero in order to find the stationary points. This can be done by factorising and finding the x values where the equation equal to zero: dy/dx=(6x-12 )(x-1). This leads to the values x =2 and x=1. After differentiating again to get the second derivative: d2y/dx2 = 12x -18. When x = 2, d2y/dx2 = 6, when x = 1, d2y/dx2 = -6. A negative second derivative indicates a maxima and a positive value indicates a minima. After substitution into the original function to find the corresponding y-values:x = 2, y = 4, Minimax = 1, y = 5, Maxima
