Abstract

Maxima and Minima Without Calculus

Professor Stanley Ogilvy has said "these are `trick methods', each applying solely to its own problem. Usually they cannot be extended, lacking the great generality of the analytic (calculus) methods." For example:

Find the coordinates where f(x) = (9x^2(sin x)^2 + 4)/(x sin x) has a maximum on (0,pi).

Niven on the other hand agrees that while calculus is good "for solving some problems in maxima and minima, the method is not universal. There are many problems that are awkward,if not impossible, to solve with elementary calculus...Thus we follow a simple maxim: If a problem can be solved more simply with calculus, leave it to calculus." Some examples:

Find the coordinates where f(x) = (9x^2(sin x)^2 + 4)/(x sin x) has a minimum on (0,\pi).

Find the quadrilateral with the largest area with a given perimeter.