## 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).