Minimum, Maximum, First and Second Derivatives of Functions

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.

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.

a graph of a functions and its derivative to explain theorem 1

Figure 1: Theorem 1: Function f and its derivative
As an example, the graph of f and its derivative f' are shown above. At the maximum (x = 2) and the minimum (x=-2) of f, f ' = 0.

Theorem 2 : (first derivative test) let f be a continuous function.
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.
Both 2.a and 2.b can clearly be verified using the graph in figure 1 above.
3.b - If f ' (a) has the same sign to the left and to the right of x = a, then f does not have
The graph in figure 2 below, shows the graph of a function f that has neither a minimum or maximum and its first derivative does not change sign.


a graph of a functions and its derivative to explain part of theorem 2

Figure 2: Theorem 2: Function f and its derivative, f ' does not change sign

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.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.
Below are shown the graph of function f, its first derivative f' and its second derivative f''. We can easily check that at the point (x = -2) where f'(-2) = 0 and f''(-2) < 0 , f has a maximum and also at the point (x = 2) where f'(2) = 0 and f'' (2)> 0 , f has a minimum at x = 2.

a graph of a functions and its derivative to explain theorem 3

Figure 3: Theorem 3: Function f, its first derivative f ' and its second derivative f ''.

More references on calculus problems
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Calculus, 11th Edition, ISBN: 978-1-118-88613-7, Jun 2016 , Howard Anton, Irl C. Bivens, Stephen Davis ,