How would you determine what sort of stationary point this curve has? x^3 - 6x^2 + 9x - 4
I would differentiate it and then turn it into an equation to find the points where the gradient equals zero. With these points at hand, I would take a second derivative, this tells me how the gradient changes with x and from this I would plug in my known points to see what value pops out. If it's postive I know that this stationary point is a minimum and if it's negative I know that this stationary point is a maximum. If the answer is zero then this hints at (but doesnt al+ways mean) a point of inflection. dy/dx = 3x2 - 12x + 9 3x2 -12x + 9 = 0 x2 - 4x + 3 = 0 (Dividing both sides with 3) (x - 3)(x-1) = 0, x=3 and x=1. d2y/dx2 = 6x -12 When x = 1, d2y/dx2 = -6 therefore a maximum When x = 3, d2y/dx2 = 6 therefore a minimum.
