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Stationary Points

Stationary points are the points on a curve where the gradient is 0. This means that at these points the curve is flat. Usually, the gradient of a curve is always changing and so the gradient is only 0 instantaneously (unless the curve is a flat line, in which case, the gradient is always 0).

Recall the graph of y=x^2. The vertex at the bottom of the curve is a STATIONARY POINT. In this case, there is a stationary point at (0,0).

Now consider the general positive quadratic in the form y=ax^2+bx+c where a\textgreater 0. There is a stationary point at the bottom of the curve; this is called a MINIMUM. Now consider a negative quadratic of the form y=ax^2+bx+c where a\textless 0. There is a stationary point at the top of the curve; this is called a MAXIMUM.

A stationary point can be found by solving \frac{dy}{dx}=0, i.e. finding the x coordinate where the gradient is 0. dy/dx is found by differentiating.

Example – Find the stationary points on the curve y=\frac{2}{3}x^3-5x^2+8x-4.

Start by solving \frac{dy}{dx}=0. \frac{dy}{dx}=2x^2-10x+8=0 i.e. x^2-5x+4=0. Factorising gives (x-4)(x-1)=0 and so the x coordinates of the stationary points are x=4 and x=1. Substituting these into the y equation gives the coordinates of the stationary points as (4,-28/3) and (1,-1/3).