My English lesson The stationary points DEFINITION A point x0, belonging to the set of definition I of a function f, is a point of relative maximum for f if there exists a neighbourhood J of x0 so that x J f(x) f(x0). A point x0, belonging to the set of definition I of a function f, is a point of relative minimum for f if there exists a neighbourhood J of x0 so that x J f(x) f(x0). DEFINITION A point x0, belonging to the definition set I of a function f, is a point of absolute maximum for f if for every x of the definition set f(x) f(x0). A point x0, belonging to the set of definition I of a function f, is a point of absolute minimum for f if for each x of the set of definition f(x) f(x0). We must emphasise that if a point is of absolute maximum (or minimum) it is also of relative maximum (or minimum). Obviously, the opposite is not true. E y C O y xH xA A B F H O xB x xC K D xD xE xK x In the graph beside, we have: relative maximum points: xA, xC, xE relative minimum points: xB, xD, xH, xK absolute maximum point: xE absolute minimum point: xH THEOREM If a function f has a relative maximum (or minimum) at a point x0 inside its set of definitions, and for x = x0 the function is derivable, then f (x0) = 0. a. y Q We already used this theorem several times, from an intuitive point of view. The theorem is not invertible: there may be points within the function s definition interval for which the derivative is zero, but which are neither maximum nor minimum points. Such are the points of horizontal inflection, as shown for F (fig. a.). P O x b. BE CAREFUL! B P Point P is an absolute minimum, but because it is angular, the derivative is not defined there. Point Q is an absolute maximum, but since it is an extreme of the interval of definition, the derivative is not defined there. 350 All points for which the derivative cancels are called stationary points. To identify the stationary points, we look for the values of the variable x at which the derivative cancels. Then, by observing the sign of the derivative, we have information on the growth or decrease of the function and we can therefore distinguish between relative maximum (first growth and then decrease), relative minimum (first decrease and then growth), horizontal flexure (always growth or always decrease). Moreover, the theorem does not exhaust all possible cases of maximum or minimum points. There may, in fact, be points of absolute maximum or minimum where the function is non-differentiable: angular points (such as point P in the figure b., which is an absolute maximum), or points at the extremes of the closed interval in which the function is defined (such as point Q in the figure, which is an absolute maximum).

My English lesson