Maxima and Minima — The First Derivative Test

A point where a function stops rising and begins to fall (or stops falling and begins to rise) is of special interest: it is where the function attains a local maximum or a local minimum. For a business, such a point may be the output that maximises profit or the output that minimises average cost, so locating these turning points is the central practical skill of this chapter.

Definitions. A function $f$ has a local maximum (or relative maximum) at $x = c$ if $f(c) \ge f(x)$ for all $x$ in some open interval around $c$ — the value $f(c)$ is at least as large as every nearby value. It has a local minimum at $x = c$ if $f(c) \le f(x)$ for all $x$ in some open interval around $c$. The value $f(c)$ itself is called the extreme value (maximum value or minimum value), and the point $\big(c, f(c)\big)$ is a turning point of the graph.

Fermat's condition. At a local maximum or minimum of a differentiable function, the tangent is horizontal, so

Thus every turning point is a critical point. The converse is not true — a critical point need not be a turning point (for instance $f(x) = x^3$ has $f'(0) = 0$, yet $x = 0$ is neither a maximum nor a minimum; it is a point of inflection). So after finding critical points we must test each one.

The First Derivative Test. Examine how the sign of $f'(x)$ changes as $x$ increases through a critical point $c$:

• if $f'(x)$ changes from positive to negative at $c$ (the function was rising, then falls), then $f$ has a local maximum at $c$;
• if $f'(x)$ changes from negative to positive at $c$ (the function was falling, then rises), then $f$ has a local minimum at $c$; …