critical point

Figure 1.6 in “Algorithms for Optimization”

A point is a critical point if ALL partial derivative of a differentiable, unbounded function are 0 OR undefined, including these subtypes:

• global optima
• local optima (local minimum, local maximum)
• inflection point

Why zero derivative? A zero derivative ensures that shifting the point by small values does not significantly affect the function value.

Univariate Calculus

The point is a critical point if

• , the domain of the function
• or does not exist

How to find a critical point:

• Find the domain of the function
• Differentiate to get first derivative and second derivative

Local Minimum

In a univariate unction

A single-variable function has a local minimum at if:

• and
• OR changes its sign from to through . This ensure that the zero first derivative occurs at the bottom of a bowl

To test whether it’s also an absolute minimum, compare with the function value at endpoints

Link to original

Local Maximum

Transclude of local-maximum#in-a-univariate-function

multivariable calculus

(Ximera OSU)

For two-variable function , a critical point occurs when *ALL of the function partial derivatives are simultaneously 0’s (or its gradient = 0). We test whether is a local extremum using the second partial derivative test

For multivariate function

Suppose has continuous second-order partial derivatives at and near a critical point in the domain . Consider its Hessian matrix

• If is positive definite, then has a local minimum at .
• Conversely, if is negative definite, then has a local maximum at
• If is indefinite, then # has a saddle point at

See motivation