Introduction to Stationary Points
When analysing the behaviour of a function, one of the most important features to identify is where the function momentarily "levels off" — neither increasing nor decreasing. These locations are called stationary points.
A point on a curve where the gradient is zero, i.e., . Stationary points include local maxima, local minima, and stationary points of inflexion.
A point where is greater than all nearby values of the function. The curve rises to this point and then falls away.
A point where is less than all nearby values of the function. The curve falls to this point and then rises again.
The word local is key here — a local maximum doesn't have to be the highest point on the entire curve (that would be a global maximum). It simply means the function is at its highest value in that neighbourhood.
Think of a mountain range: a local maximum is like any individual mountain peak — it's higher than its immediate surroundings, but there may be a taller peak elsewhere in the range.
