费马引理

需要注意的是，费马引理仅仅给出了函数在某个点为极值的必要条件。也就是说，有些驻点不是极值，它们是拐点。要想知道一个驻点是不是极值，并进一步区分最大值和最小值，我们需要分析二阶导数（如果它存在）。

定理

设函数f(x)在点x0的某邻域U(x0)内有定义，并且在x0处可导，如果对任意的 x∈ U(x₀)，有 f(x)≤ f(x₀) 或 f(x)≥ f(x₀)

那么 f^(′)(x₀)=0。

费马引理的一个推论是，函数f在定义域A内的最大值和最小值只能在边界上，不可导的点，或驻点取得。

证明

假设 x₀ 是一个极大值（如果 x₀ 是极小值，证明亦类似）。那么存在一个 δ >0，使得对于所有的 |x-x₀|<δ，都有 f(x₀)≥ f(x)。因此对于任何 h∈ (0,δ)，有： （公式略）

由于当 h 从上方趋于0时，这个比值的极限存在且为 f'(x₀)，我们便有 f'(x₀)≤ 0。另一方面，当 h∈ (-δ,0) 时，我们注意到： （公式略）

当 h 从下方趋于0时，这个极限存在，且等于 f'(x₀)，我们又有 f'(x₀)≥ 0。

因此 f'(x₀)=0。

In calculus and real analysis, the interior extremum theorem states that any local extremum of a real function at which it is differentiable is a stationary point. It is also known as Fermat's theorem, named after the French mathematician Pierre de Fermat.

The interior extremum theorem gives a necessary, but not sufficient condition for local extrema at which the function is differentiable, as some stationary points are not local extrema.

The second derivative, if non-zero, can be used to determine whether a local extremum at which the function is twice differentiable is a maximum or a minimum. However, the second derivative can be zero at local extrema.

History

Pierre de Fermat proposed in a collection of treatises titled Maxima et minima a method to find maximum or minimum, similar to the modern interior extremum theorem using an approach he called adequality. After Marin Mersenne passed the treatises onto René Descartes, Descartes was doubtful, remarking "if [...] he speaks of wanting to send you still more papers, I beg of you to ask him to think them out more carefully than those preceding". Descartes later agreed that the method was valid.

Statement

One way to state the interior extremum theorem is that, if a function has a local extremum at some point and is differentiable there, then the function's derivative at that point must be zero. In precise mathematical language:

Let f: (a,b)→ R be a function from an open interval ⁠ (a,b) ⁠ to ⁠ R ⁠, and suppose that x₀∈ (a,b) is a point where f has a local extremum. If f is differentiable at x₀, then f'(x₀)=0.

Another way to understand the theorem is via the contrapositive statement: if the derivative of a function at any point is not zero, then there is not a local extremum at that point. Formally:

If f is differentiable at x₀∈ (a,b), and f'(x₀)≠ 0, then x₀ is not a local extremum of f.

Corollary

Every global extremum of a function f on a domain A occurs only at the boundary of A, non-differentiable points, or stationary points.

• If x₀ is a global extremum of f, then one of the following is true:
• boundary: x₀ is in the boundary of A
• non-differentiable: f is not differentiable at x₀
• stationary point: f'(x₀)=0

The function ⁠ f(x)=x³ ⁠ has no extrema. The function ⁠ f(x)=x³-x ⁠ has no global extrema, although it has local extrema.

Extension

A similar statement holds for the partial derivatives of multivariate functions. Suppose that some real-valued function of the real numbers f=f(t₁,t₂,…,tₖ) has an extremum at a point C, defined by C=(a₁,a₂,…,aₖ). If f is differentiable at C, then: （公式略） for each i=1,2,…,k.

The statement can also be extended to differentiable manifolds. If f:M→ R is a differentiable function on a manifold M, then its local extrema must be critical points of f, in particular points where the exterior derivative df is zero.

Applications

The interior extremum theorem is central for determining maxima and minima of piecewise differentiable functions of one variable: an extremum is either a stationary point (that is, a zero of the derivative), a non-differentiable point (that is a point where the function is not differentiable), or a boundary point of the domain of the function. Since the number of these points is typically finite, the computation of the values of the function at these points provides the maximum and the minimum, simply by comparing the obtained values.

Proof

Suppose that x₀ is a local maximum. (A similar argument applies if x₀ is a local minimum.) Then there is some neighbourhood around x₀ such that f(x₀)≥ f(x) for all x within that neighborhood. If x>x₀, then the difference quotient （公式略） is non-positive for x in this neighborhood. This implies （公式略）

Similarly, if x<x₀, then the difference quotient is non-negative, and so （公式略）

Since f is differentiable, the above limits must both be equal to f'(x₀). This is only possible if both limits are equal to 0, so f'(x₀)=0.