SL 5.8—Testing for max and min, optimisation. Points of inflexion

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.

Stationary Point: A point on a curve where the gradient is zero, i.e., $f^{\prime }(x)=0$. Stationary points include local maxima, local minima, and stationary points of inflexion.

Local Maximum: A point where $f(a)$ is greater than all nearby values of the function. The curve rises to this point and then falls away.

Local Minimum: A point where $f(a)$ 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.