A curve is given by the equation y = (1/3)x^3 -4x^2 +12x -19. Find the co-ordinates of any stationary points and determine whether they are maximum or minimun points.
Stationary points - dy/dx =0dy/dx = x2 -8x +12 ( differentiating- early a level skill). Solving x2 -8x +12=0 gives x=6, x=2 (solving quadratics- gcse skill). Subbing these values back into the original equation gives the co-ordinates of the stationary points- (2,-8), (6,-19)To determine whether they are maximum or minimum points we need to find the second derivative (differentiation)d2y/dx2 = 2x-8.Subbing in our x co-ordinate for the stationary point (2,-8) gives d2y/dx2= -4. Since this is negative, (2,-8) is a maximum point. Similarly, subbing in the x value of our other stationary point gives d2y/dx2= 4. Since this is positive, (6,-19) is a minimum point.
