how to find absolute extrema

2.4 Absolute Extrema

In this section, we learn how to find the largest and smallest \\(y\\) values (_absolute extrema_) on the graph. + We examine only the critical values and the endpoints of the interval.

Idea

The largest \(y\)-value on the graph is the absolute maximum, and the smallest \(y\)-value on the graph is the absolute minimum.

Definition: Absolute Extrema

Suppose \( f \) is a function on the interval \( [a,b] \). Then,

\( \bullet \) The \( y \)-value \( f(c) \) is an \( \bf{\text{absolute maximum}} \) if \( f(c) \geq f(x) \) on \( [a,b] \).
\( \bullet \) The \( y \)-value \( f(c) \) is an \( \bf{\text{absolute minimum}} \) if \( f(c) \leq f(x) \) on \( [a,b] \).

Procedure

Maximum-Minimum Principle

Suppose that \( f \) is a continuous function over the closed interval \( [a,b] \). To find the absolute extrema (maxima and minima) over \( [a,b] \), do the following:

\( (a) \) Find \(f'(x)\)
\( (b) \) Then, determine all critical values of \( f \) in \( (a,b) \), i.e. all \(c\) such that

\begin{align} f'(c) &= 0 \quad \text{or} \quad f'(c) = DNE \end{align}

\( (c) \) List all values from the previous step and the endpoints:

\begin{align} a, c_1, c_2, \dots, c_n, b \end{align}

\( (d) \) Evaluate \( f(x) \) for each value in step \( (c) \):

\begin{align} f(a), f(c_1), f(c_2), \dots, f(c_n), f(b) \end{align}

The largest of these values is the absolute maximum of \(f\) on \( [a,b] \), and the smallest of these values is the minimum.

Examples

Example 1

Find the absolute maximum and minimum for

\begin{align} f(x) &= -x^3 -3x^2 + 9x + 12 \quad \text{over} \; \; [-4,2] \end{align}

Solution:
\( (a) \) Find \( f'(x) \)

\begin{align} f'(x) &= -3x^2 -6x +9 \end{align}

\( (b) \) Determine all critical values:

From previous classes, we have: \( x=-3, 1 \).

\( (c) \) Points to test:

\begin{align} -4, -3, 1, 2 \end{align}

\( (d) \) Evaluate the original function at points in previous step:

x-value	\(f\)
-4	\(f(-4) = -8\)
-3	\(f(-3) = -15\)
1	\(f(1) = 17\)
2	\(f(2) = 10\)

From the table we observe:

The absolute max is \(17\) and it occurs at \(x=1\)
The absolute min is \(-15\) and it occurs at \(x=-3\)

Example 2

Find the absolute max and min for

\begin{align} f(x) &= x + \frac{4}{x} \quad \text{on} \; [-8,-1] \end{align}

Solution:
\( (a) \) Find \( f'(x) \)

\begin{align} f'(x) &= 1- \frac{4}{x^2} \end{align}

\( (b) \) Determine all critical values:

Solve \( f'(x) =0 \):

\begin{align} f'(x) = 1- \frac{4}{x^2} &= 0 \\ 1 &= \frac{4}{x^2} \\ x^2 &= 4 \\ x& = \pm 2 \end{align}

In addition, \( f'(x) \) is not defined at \( x=0 \) but neither is the original function, so it is not a critical value.

\( (c) \) The interval of consideration is \( [-8,-1] \), and any critical values outside this interval are ignored. Here, we ignore \( x=2 \).

Points to test:

\begin{align} -8, -2, -1
\end{align}

\( (d) \) Evaluate the original function at points in previous step:

x-value	\(f\) original func.
-8	\(f(-8) = -8.5\)
-2	\(f(-2) = -4\)
-1	\(f(-1) = -5\)

From the table we observe:

The absolute max is \(-4\) and it occurs at \(x=-2\)
The absolute min is \(-8.5\) and it occurs at \(x=-8\)