A local extremum is of little significance: generally, only the global extremum matters. Yet some authors merely set the derivative of a function of one variable equal to zero and claim to have found the desired global extremum (which is not enough to prove it).
There are, however, situations in which the distinction between local and global extrema disappears. The following result, dubbed the carefree theorem by Professor François Jongmans (Université de Liège), says that if the real-valued function f, defined and continuous on the interval I, has only one local extremum in the interior of I, then it is a strict global maximum or minimum, respectively, for f on I, while an extremum of the opposite type, if it exists, can occur only at an endpoint of I.
Consider the cubic function defined by f(x) = 3*x 2 – *x 3. A local analysis yields a local maximum (with value 4, attained at x = 2) and a local minimum (with value 0 at x = 0). The global analysis, however, depends heavily on the interval I under consideration! The carefree theorem tells us that the global maximum of f on [1, 3] is 4 and is attained at the sole local maximum lying in I, while the minimum is 2 and is attained at the lower endpoint 1 of I. By contrast, if we widen the interval under consideration, the maximum of f soon exceeds 4…
Unfortunately, the carefree theorem does not extend to functions of several variables. We must instead use properties arising from the convexity (or concavity) of the function under study; under certain assumptions, these again allow us to pass automatically from local to global.
