Maximum and Minimum of a Function

The maximum and minimum of a function are points within an interval $ x \in [a, b] $ at which the function $ y = f(x) $ attains its greatest and least values, respectively.

These extrema may be either absolute or relative:

Absolute maximum or minimum: The point at which $ f(x) $ achieves its largest or smallest value over the entire interval $ [a, b] $.
Relative (local) maximum or minimum: A point where $ f(x) $ achieves a largest or smallest value within some neighborhood of $ x_0 $.

Example

example of absolute and relative maximum and minimum

At $ x_1 $, the function $ f(x) $ attains an absolute maximum on $ [a, b] $, and at $ x_2 $ it attains an absolute minimum.

At $ x_3 $ and $ x_4 $, the function has only a relative (local) maximum and minimum, respectively.

Note: An absolute maximum (or minimum) on $ [a, b] $ is, by definition, also a relative maximum (or minimum) in any neighborhood of that point. Thus, $ x_1 $ is both an absolute and a relative maximum, while $ x_2 $ is both an absolute and a relative minimum.

Absolute Maxima and Minima
Relative Maximum and Minimum
Notes

Absolute Maxima and Minima

Absolute Maximum

Let $ y=f(x) $ be a function defined on an interval $ I $. The value $ M=f(x_0) $, where $ x_0 \in I $, is called the absolute maximum of the function on $ I $ if \[ f(x_0) \ge f(x) \qquad \forall x \in I \]

Simply put, $ f(x_0) $ is the absolute maximum if it is the largest value the function takes anywhere on the interval $ I $.

If an absolute maximum exists, it is represented on the Cartesian graph by the point $ (x_0,M) $, where $ M=f(x_0) $.

Absolute Minimum

Let $ y=f(x) $ be a function defined on an interval $ I $. The value $ m=f(x_0) $, where $ x_0 \in I $, is called the absolute minimum of the function on $ I $ if \[ f(x_0) \le f(x) \qquad \forall x \in I \]

Likewise, $ f(x_0) $ is the absolute minimum if it is the smallest value the function takes anywhere on the interval $ I $.

If an absolute minimum exists, it is represented on the Cartesian plane by the point $ (x_0,m) $, where $ m=f(x_0) $.

Note. Whether a function has an absolute maximum or an absolute minimum depends both on the function itself and on the domain over which it is defined.

Example

Consider the function

\[ y=x^2+1 \]

Over its entire domain $ I=\mathbb{R} $, the function has an absolute minimum at $ x=0 $, because

\[ f(0)=1 \]

However, it does not have an absolute maximum. As the absolute value of $ x $ increases, the function grows without bound.

Example of a function with an absolute minimum and no absolute maximum

Now restrict the domain to the closed interval $ I=[1,3] $. On this interval, the function has both an absolute minimum and an absolute maximum.

Since the function is increasing for every $ x \ge 0 $, the smallest value occurs at the left endpoint of the interval, while the largest value occurs at the right endpoint:

\[ f(1)=2 \qquad \text{(absolute minimum)} \] \[ f(3)=10 \qquad \text{(absolute maximum)} \]

Therefore, on the interval $ [1,3] $, the absolute minimum is $2$ and the absolute maximum is $10$.

Example of an absolute minimum and an absolute maximum on a closed interval

Relative Maximum and Minimum

Consider a point $$ x_0 $$ and examine all points within a neighborhood of radius $ \delta $ about $ x_0 $:

$$ |x - x_0| < \delta $$

If $ f(x_0) $ exceeds all nearby values of $ f(x) $, then $ x_0 $ is a relative maximum:

$$ f(x_0) > f(x) $$

For example:

example of a relative maximum

Conversely, if $ f(x_0) $ is smaller than all nearby values of $ f(x) $, then $ x_0 $ is a relative minimum:

$$ f(x_0) < f(x) $$

For example:

example of a relative minimum

Note: For relative extrema, the inequality (greater than or less than) holds only within some neighborhood of $x_0 $. It need not hold over the entire interval $ [a, b] $ where the function is defined.

Notes

The following theorem is one of the most important results concerning absolute maxima and minima.

Extreme Value Theorem
If a function $ f(x) $ is continuous on a closed and bounded interval $ [a,b] $, then it always attains both an absolute maximum and an absolute minimum on that interval. \[ \exists \, x_M, x_m \in [a,b] \; \text{such that} \; f(x_m) \le f(x) \le f(x_M) \qquad \forall x \in [a,b] \] In other words, there are points $ x_m $ and $ x_M $ in the interval where the function reaches its smallest and largest values, respectively.

And so on.