A curve has equation: y = x^3 - 3x^2 + 5. Show that the curve has a minimum point when x = 2.

A minimum point will have a gradient of 0 (although so will a maximum point or a point of inflection). dy/dx = 3x²-6x. We can substitute x = 2 into this equation to give 0 (alternatively solve 3x²-6x = 0 to give x = 2 as a root). Thus the gradient at x = 2 is 0 so it is either a minimum point, maximum point or point of inflection. To prove it is in fact a minimum point, we can differentiate the original equation of the curve once more to give d²y/dx²= 6x - 6 (the second derivative). Substituting x = 2 into this equation gives a positive value. Positive value = minimum point; negative value = maximum point; zero value requires further investigation by finding gradients of x values slightly smaller and larger than 2 (this is also a possible alternative method to determine which type of stationary point it is to begin with).