Fermat's Theorem

A point x₀ is caleld a stationary point (Critical point/value), if f'(x₀) = 0. Also the value of the function f(x₀) is said to be stationary (Critical) at that point. The maximum and minimum values of a function are called extreme values. We shall also be interested in local extrema of f which are defined as follows

f(c) is a local maximum of f if there exists an open interval (a,b) containing c such that f(x) ≤ f(c) for all x in (a,b);
f(c) is a local minimum of f if there exists an open interval (a,b) containing c such that f(x) ≥ f(c) for all x in (a,b);

The next theorem shows that a local extreme point in the interior of the domain of a differentiable function must be a stationary point. This theorem is named for French mathematician Pierre de Fermat (1601– 1665). Although his life predates the discovery of calculus proper, Fermat computed tangent lines and extrema for many families of curves.

punti di estremo — **Fig. 1**. If f has extreme points inside the interval in which is differentiable, the derivate is null at these points.

Theorem 5.4.1. Fermat's Theorem. Let f: [a,b] ⟶ ℝ, differentiable in c ∈ (a,b). If c is a local extremum point then f'(c) = 0

Proof. Suppose, f has in c a local maximum (same for minimum). Then f(c) ≥ f(x) is x is sufficiently close to c. This implies that if h is sufficiently close to 0, with h being positive or negative, then

f(c) ≥ f(c + h)

and therefore

f(c + h) − f(c) ≤ 0

We can divide both sides of an inequality by a positive number. Thus, if h > 0 and h is sufficiently small, we have

[f(c + h) − f(c)] / h ≤ 0

Taking the right-hand limit and the left-hand limit, we get

f_{+}^{'} (c) = lim_{h \to 0^{+}} \frac{f (c + h) - f (c)}{h} \leq 0 and f_{-}^{'} (c) = lim_{h \to 0^{-}} \frac{f (c + h) - f (c)}{h} \geq 0

Since the function is derivable in c then f'(c) = f'₋(c) = f'₊(c) = 0. □

We must be careful when using Fermat’s Theorem. The converse of Fermat’s Theorem is false in general: Even when f'(x₀) = 0 there need not be a maximum or minimum at x₀ as in the following case in which f has a point of inflection at x = 0:

f(x) = x^3 — The function f(x) = x³ has a stationary point at x = 0 but is strictly monotone on (−∞, ∞).

The function f(x) = x³ has derivative f'(x) = 3x² which is zero in the origin.

x local extreme values ⇒ x stationary point values

but the converse implication is not generally true.

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