give the coordinates of the stationary points of the curve y = x^4 - 4x^3 + 27 and state with reason if they are minumum, maximum, or points of inflection.

differentiating the equation gives dy/dx = 4x^3 - 12x^2 dy/dx = 4x^2(x - 3)

at a turning point, dy/dx = 0. Solving the equation 4x^2(x - 3) = 0 yeilds x = 0, x = 3

putting 0 and 3 back into the curves equation gives y = 27 when x = 0 y = 0 when x = 3

The coordinates are therefore (0,27) and (3,0).

To find out the nature of the turning points we must find the second derivative, d^2y/dx^2.

d^2y/dx^2 = 12x^2 - 24x

inputting the x values of the turning points gives us d^2y/dx^2 = 0 for (0,27), this is neither +ve or -ve, so the point is a point of inflection, d^2y/dx^2 = 36 for (3,0), this is positive, indicating an increase in gradient, so the the point is a minimum.