# Minimum, Maximum, First and Second Derivatives

A tutorial on how to use calculus theorems using first and second derivatives to determine whether a function has a relative maximum or minimum or neither at a given point. These are some of the most important theorems in problem solving. This tutorial uses an interactive applet.

 Theorem 1 : If function f has a relative minimum or maximum at x = a, we either have f ' (a) = 0 (stationary point) or f ' (a) does not exist. An applet that displays the graphs of f, f ' and f '' may be used to fully understand the theorem. Start the applet by pressing the button "press here to start". Your browser is completely ignoring the tag! In this applet you may expand and reflect the graph of f vertically using the button "change height" and also shift the graph horizontally "change position". Verify that whenever there is a relative minimum or maximum at a certain point, f '(x) is equal to zero at that point (stationary point). Theorem 2 : (first derivative test) let f be a continuous function. 3 possible cases a, b and c. 2.a - If f ' (a) = 0 or f ' (x) does not exists at x = a and if f ' (x) < 0 to the left of a and f ' (a) > 0 to the right of a, then f has a relative minimum at x = a. 2.b - If f ' (a) = 0 or f ' (x) does not exists at x = a and if f ' (x) > 0 to the left of a and f ' (a) < 0 to the right of a, then f has a relative maximum at x = a. 3.b - If f ' (a) has the same sign to the left and to the right of x = a, then f does not have relative minimum or maximum (relative extremum) at x = a. Use the applet to set the height so that the f has relative minimum and check that f ' (x) has the required signs as stated above in case a). Use the applet to set the height so that the f has relative maximum and check that f ' (x) has the required signs as stated above in case b). Theorem 3 : (first and second derivatives test) Suppose that both f ' and f '' exists at x = a and that f ' (a) = 0 (stationary point). 3 possible cases a, b and c. 3.a - If f '' (a) > 0 , f has a relative minimum at x = a. 3.b - If f '' (a) < 0 , f has a relative maximum at x = a. 3.c - If f '' (a) = 0 , no conclusion can be made, use theorem 2 above. Use the applet to set the height so that f has relative minimum and check that f ''(x) > 0 at the point where the relative maximum is positioned. Use the applet to set the height so that the f has relative maximum and check that f ''(x) < 0 at the point where the relative minimum is positioned. More references on calculus problems

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Updated: 27 November 2007 (A Dendane)