Find the Co-ordinates and nature of all stationary points on the curve y=x^3 - 27x, and attempt to sketch the curve

Firstly we need to find the stationary points, we know that when the first derivative of the curve is equal to 0, this means it is a stationary point as there is no effective gradient.
The first derivative is 3x^2 - 27. If we equal this to 0 and solve this we get (x-3)(x+3)=0. This means that there are two stationary points, one at x=3, and one at x=-3. If we then substitute these x values into the initial curve equation we can determine the corresponding y values, which are -54 and 54 respectively. This means that the two stationary points are (-3, 54) and (3, -54).
Now if want to determine the nature of the stationary points we look to the second derivative which is 6x. Once the x value is substituted in, a positive second value denotes a minimum point, a negative value denotes a positive, and a value of 0 denotes a point of substitution. We can find that the point of (-3,54) is a a local maxima, and (3, -54) is a local mimima.