Stationary Points

A stationary point is a point in a function where the first derivative is zero.
There are three types of stationary points: local maximum, local minimum, and inflexion point.

Local Maximum and Minimum

A point is a local maximum if the functions values just to the left and right are less than the function’s value at that point.
A local minimum has function values to the left and right that are greater than its own.
To locate these points, set the first derivative of the function to zero and solve for the variable.

An inflexion point is a point where the function changes concavity.
That means it switches from being concave up (shaped like a U) to concave down (shaped like an n), or vice versa.
To find inflexion points, set the second derivative of the function equal to zero and solve for the variable.

The second derivative test can be used to determine whether a stationary point is a maximum, minimum, or inflexion point.
If the second derivative at the point is positive, the point is a local minimum.
If the second derivative at the point is negative, the point is a local maximum.
If the second derivative at the point is zero, the test is inconclusive, and the point could be a local maximum, local minimum, or an inflexion point. Use other methods like the first derivative test or graphical methods to determine the nature of the stationary point.