Interior extremum theorem

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\colon (a,b) \rightarrow \mathbb{R} be a function from an open interval to , and suppose that x_0 \in (a,b) is a point where f has a local extremum. If f is differentiable at x_0, then f'(x_0) = 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 x_0 \in (a,b), and f'(x_0) \neq 0, then x_0 is not a local extremum of f. Corollary The global extrema of a function f on a domain A occur only at boundaries, non-differentiable points, and stationary points. If x_0 is a global extremum of f, then one of the following is true: • boundary: x_0 is in the boundary of A • non-differentiable: f is not differentiable at x_0 • stationary point: x_0 is a stationary point of f Extension A similar statement holds for the partial derivatives of multivariate functions. Suppose that some real-valued function of the real numbers f = f(t_1, t_2, \ldots,t_k) has an extremum at a point C, defined by C = (a_1, a_2,\ldots ,a_k). If f is differentiable at C, then:\frac{\partial}{\partial t_i}f(a_i)=0where i = 1, 2, \ldots ,k. The statement can also be extended to differentiable manifolds. If f : M \to \mathbb{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 x_0 is a local maximum. (A similar argument applies if x_0 is a local minimum.) Then there is some neighbourhood around x_0 such that f(x_0) \ge f(x) for all x within that neighborhood. If x > x_0, then the difference quotient \frac{f(x) - f(x_0)}{x - x_0} is non-positive for x in this neighborhood. This implies \lim_{x\rightarrow x_0^+}\frac{f(x) - f(x_0)}{x - x_0} \le 0. Similarly, if x , then the difference quotient is non-negative, and so \lim_{x\rightarrow x_0^-}\frac{f(x) - f(x_0)}{x - x_0} \geq 0. Since f is differentiable, the above limits must both be equal to f'(x_0). This is only possible if both limits are equal to 0, so f'(x_0) = 0. == See also ==