Question

Sketch the graph of \(f\left( x \right) = \frac{{\bf{1}}}{x}\), \({\bf{1}} < x < {\bf{3}}\) by hand and use your sketch to find the absolute and local maximum and minimum values of \(f\).

Short answer

The graph has no absolute minimum, the graph has no absolute maximum, the graph has no local maximum or local minimum.

Step-by-step solution

Definition for an absolute minimum and maximum

For any number \(c\) in the domain \(D\) of a function \(f\), \(f\left( c \right)\) is said to be the absolute minimum value of \(f\) on \(D\) if \(f\left( c \right) \le f\left( x \right)\) for all \(x\) in \(D\).

For any number \(c\) in the domain \(D\) of a function \(f\), \(f\left( c \right)\) is said to be the absolute maximumvalue of \(f\) on \(D\) if \(f\left( c \right) \ge f\left( x \right)\) for all \(x\) in \(D\).

Definition for a local minimum and maximum

For any number \(c\) in the domain \(D\) of a function \(f\), \(f\left( c \right)\) is said to be the local minimumvalue of \(f\) if \(f\left( c \right) \le f\left( x \right)\) when \(x\) is near \(c\).

For any number \(c\) in the domain \(D\) of a function \(f\), \(f\left( c \right)\) is said to be the local maximum value of \(f\) if \(f\left( c \right) \ge f\left( x \right)\) when \(x\) is near \(c\).

Find the absolute minimum and maximum value of the function

Consider a graph of the function \(f\left( x \right) = \frac{1}{x}\) for \(1 < x < 3\) as follows.

By the definition of absolute minimum, the graph does not contain absolute minimum.

By the definition of absolute maximum, the graph does not contain absolute maximum.

Find the local minimum and maximum value of the function

The point \(f\left( 2 \right)\) in the graph is the minimum value in the smallest neighborhood of \(f\left( 2 \right)\) because \(f\left( 2 \right) = 0.5 \le f\left( x \right)\) for all \(x\) nearest to \(2\).

By the definition of local minimum, there is no local minimum.

By the definition of local maximum, the graph does not contain local maximum.

Thus, the graph has no absolute minimum, and the graph has no absolute maximum. The graph has no local maximum or local minimum.