How Derivatives Find Global and Local Optima (Minima & Maxima)

1 Global Optimal Point

For a function, a global minimum point is a point where the function attains its smallest value over the entire domain.

2 Local Optimal Point

A local minimum point is a point where the function’s value is the smallest within a neighborhood around that point. Every global minimum is also a local minimum.

Local and global maxima are defined similarly by replacing "minimum" with "maximum".

The relationship between global and local extrema is illustrated in the figure below.

Critical Points

When a function is smooth (differentiable everywhere), a critical point is where its derivative equals zero, meaning the tangent line is horizontal.

If a point is a critical point, it is either a maximum, a minimum, or a saddle point, as shown in the figure.

Near a saddle point, the derivative’s magnitude decreases to zero and then the function continues to increase or decrease in the same direction, unlike at maxima or minima where the direction reverses.

Finding Critical Points

To locate minima or maxima, solve for critical points by setting the derivative equal to zero.

Example 1

For a quadratic function, its derivative is linear, yielding a single critical point. This point is both the local and global extremum.

Example 2

For a cubic function, the derivative is quadratic, leading to two critical points. Solving the quadratic equation gives the locations of these points, which can be classified as local minima, maxima, or saddle points.