Maxima and Minima

Beyond knowing where a function rises or falls, it is often just as important to know where it reaches its highest or lowest value overall — its extreme values.

Let $y = f(x)$ be a real function defined on a set $D$. The value $f(c)$, for some $c \in D$, is called the absolute minimum value of $f$ on $D$ if $f(x) \geq f(c)$ for every $x \in D$ — no value the function takes anywhere else in $D$ is smaller. Likewise, $f(c)$ is the absolute maximum value of $f$ on $D$ if $f(x) \leq f(c)$ for every $x \in D$ — no other value is larger.

Whenever $f$ attains either its absolute maximum or its absolute minimum at some point $c$ in its domain, $f$ is said to have an absolute extremum at $c$; the value $f(c)$ itself is called an absolute extremum value, and $c$ is called a point of extremum. Locating these points — and telling maxima apart from minima — is what the rest of this chapter's treatment of maxima and minima builds towards, using exactly the derivative tools (critical points, increasing and decreasing behaviour) developed in the previous two sections.

Local maxima and minima

Absolute extrema describe the single highest and lowest values over the whole domain. Very often, though, a function has several "peaks" and "valleys" — points that are highest or lowest only when compared with their immediate surroundings. On the graph of a continuous function these show up as peak-type points (the curve rises to them, then falls) and valley-type points (the curve falls to them, then rises). A point $c$ is a local maximum if the curve is increasing just to its left and decreasing just to its right, so $f(c)$ is the largest value in a small neighbourhood; it is a local minimum if the curve is decreasing just to its left and increasing just to its right. The tangent at every such peak or valley is parallel to the $x$-axis, so its slope is zero — these are critical points. But not every critical point is an extremum: $h(x)=x^3$ has $h'(0)=0$ and yet $x=0$ is neither a maximum nor a minimum, because the function keeps increasing on both sides (such a point is a point of inflexion).

Formally, let $c$ be an interior point of the domain of $f$. Then:

First derivative test

Since the sign of $f'$ records whether the function is rising or falling, a critical point $c$ (at which $f$ is continuous) can be classified by watching how $f'$ changes sign as $x$ passes through it:

In words, a sign change from decreasing to increasing marks a valley, a change from increasing to decreasing marks a peak, and no change at all marks an inflexion.

Second derivative test

When the second derivative is easy to compute it gives a quicker test at a critical point $c$ where $f'(c)=0$:

If $f''(c)=0$ the second derivative test fails, and one falls back on the first derivative test to decide whether $c$ is a local maximum, a local minimum, or a point of inflexion.

Maximum and minimum values in a closed interval …