Stationary Point

Tags

math

A point on a curve where the gradient of the curve is 0. These points manifest themselves on curves that demonstrate both increasing and decreasing behaviour. When the curve reaches a local minimum or maximum its gradient drops to 0 as it flips to an increasing gradient (from a decreasing one) or vice versa.

\begin{figure}
  \centering
  \begin{tikzpicture}
    \begin{axis}[axis lines=center, xmin=-5, xmax=5, ymin=-5, ymax=5, domain=-5:5, grid=both]
      \addplot[color=green]{x^3 + 3*x^2};,
      \addplot[only marks] coordinates {(-2, 4) (0, 0)};,
    \end{axis}
  \end{tikzpicture}
  \qquad
  \caption{A graph with 2 stationary points.}
  \label{fig:stationary-point-demo}
\end{figure}

You can find the stationary points of a function \( y \) by finding the roots of the derivative of that function \( \frac{\,dy}{\,dx} = 0 \).

To assert the kind of stationary point, whether it's a local minimum (before the gradient starts increasing again) or a local maximum (where the curve reaches a maximum before the gradient starts decreasing again) you can use the second derivative of \( y \), \( \frac{\,d^2 y}{\,dx^2} \) and substitute the value of \( x \) of the stationary point. If:

• \( \frac{\,d^2 y}{\,dx^2} > 0 \) then you've found a minimum point.
• \( \frac{\,d^2 y}{\,dx^2} < 0 \) then you've found a maximum point.
• \( \frac{\,d^2 y}{\,dx^2} = 0 \) then we don't know, further examination is needed.