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".

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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)