Editing Differential calculus (section)

=== Optimization ===
If {{math|''f''}} is a [[differentiable function]] on {{math|ℝ}} (or an [[open interval]]) and {{math|''x''}} is a [[local maximum]] or a [[local minimum]] of {{math|''f''}}, then the derivative of {{math|''f''}} at {{math|''x''}} is zero. Points where {{math|''f<nowiki>'</nowiki>''(''x'') {{=}} 0}} are called ''[[critical point (mathematics)|critical points]]'' or ''[[stationary point]]s'' (and the value of {{math|''f''}} at {{math|''x''}} is called a '''critical value'''). If {{math|''f''}} is not assumed to be everywhere differentiable, then points at which it fails to be differentiable are also designated critical points.

If {{math|''f''}} is twice differentiable, then conversely, a critical point {{math|''x''}} of {{math|''f''}} can be analysed by considering the [[second derivative]] of {{math|''f''}} at {{math|''x''}} :
* if it is positive, {{math|''x''}} is a local minimum;
* if it is negative, {{math|''x''}} is a local maximum;
* if it is zero, then {{math|''x''}} could be a local minimum, a local maximum, or neither. (For example, {{math|''f''(''x'') {{=}} ''x''<sup>3</sup>}} has a critical point at {{math|''x'' {{=}} 0}}, but it has neither a maximum nor a minimum there, whereas {{math|''f''(''x'') {{=}} ± ''x''<sup>4</sup>}} has a critical point at {{math|''x'' {{=}} 0}} and a minimum and a maximum, respectively, there.)
This is called the [[second derivative test]]. An alternative approach, called the [[first derivative test]], involves considering the sign of the {{math|''f<nowiki>'</nowiki>''}} on each side of the critical point.

Taking derivatives and solving for critical points is therefore often a simple way to find local minima or maxima, which can be useful in [[Optimization (mathematics)|optimization]]. By the [[extreme value theorem]], a continuous function on a [[closed interval]] must attain its minimum and maximum values at least once. If the function is differentiable, the minima and maxima can only occur at critical points or endpoints.

This also has applications in graph sketching: once the local minima and maxima of a differentiable function have been found, a rough plot of the graph can be obtained from the observation that it will be either increasing or decreasing between critical points.

In [[higher dimension]]s, a critical point of a [[Scalar (mathematics)|scalar value]]d function is a point at which the [[gradient]] is zero. The [[second derivative]] test can still be used to analyse critical points by considering the [[eigenvalue]]s of the [[Hessian matrix]] of second partial derivatives of the function at the critical point. If all of the eigenvalues are positive, then the point is a local minimum; if all are negative, it is a local maximum. If there are some positive and some negative eigenvalues, then the critical point is called a "[[saddle point]]", and if none of these cases hold (i.e., some of the eigenvalues are zero) then the test is considered to be inconclusive.

==== Calculus of variations ====
{{Main|Calculus of variations}}

One example of an optimization problem is: Find the shortest curve between two points on a surface, assuming that the curve must also lie on the surface. If the surface is a plane, then the shortest curve is a line. But if the surface is, for example, egg-shaped, then the [[Shortest path problem|shortest path]] is not immediately clear. These paths are called [[geodesic]]s, and one of the most fundamental problems in the calculus of variations is finding geodesics. Another example is: Find the smallest area surface filling in a closed curve in space. This surface is called a [[minimal surface]] and it, too, can be found using the calculus of variations.