A curve has equation y = 20x −x2 −2x3 . (A) Find the x-coordinates of the stationary points of the curve.
Firstly, differentiate the equation to find dy/dx.
dy/dx = 20 - 2x - 6x2
As dy/dx represents the gradient, we know that for a stationary point the gradient must be zero, hence for the stationary points, we set dy/dx = 0.
dy/dx = 20 - 2x - 6x2 = 0
Now, we have a quadratic equation, which we can now put into brackets to find our solutions.
dy/dx = 0 = 6x2 - 2x +20 = (x+2)(6x-10)
From these brackets, we know if one set were to be zero, dy/dx would be zero and we will find our x coordinates for our stationary points.
If (x+2) = 0, then x=-2
Or, if (6x-10) = 0, then x=10/6 = 5/3 simplified
