Question

Use the graph to state the absolute and local maximum and minimum values of the function.

Short answer

The graph has no absolute minimum, the absolute maximum is at \(f\left( 4 \right) = 5\), the local maximum is at \(f\left( 4 \right) = 5\), and \(f\left( 6 \right) = 4\), and the local minimum is at \(f\left( 1 \right) = 3\), \(f\left( 2 \right) = 2\) and \(f\left( 5 \right) = 3\).

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 maximum value 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 \(f\) as follows.

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

The point \(f\left( 4 \right)\) in the graph is the maximum value in \(f\left( {{\mathbb{R}^ + }} \right)\) where \({\mathbb{R}^ + }\) is the domain because \(f\left( 4 \right) = 5 \ge f\left( x \right)\) for all \(x\) in \({\mathbb{R}^ + }\).

By the definition of absolute maximum, \(f\left( 4 \right) = 5\) is the absolute maximum.

Find the local minimum and maximum value of the function

The point \(f\left( 1 \right)\) in the graph is the minimum value in the smallest neighborhood of \(f\left( 1 \right)\) because \(f\left( 1 \right) = 3 \le f\left( x \right)\) for all \(x\) nearest to \(1\).

By the definition of local minimum, \(f\left( 1 \right) = 3\) is the local minimum.

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

By the definition of local minimum, \(f\left( 2 \right) = 2\) is the local minimum.

The point \(f\left( 5 \right)\) in the graph is the minimum value in the smallest neighborhood of \(f\left( 5 \right)\) because \(f\left( 5 \right) = 3 \le f\left( x \right)\) for all \(x\) nearest to \(5\).

By the definition of local minimum \(f\left( 5 \right) = 3\) is the local minimum.

The point \(f\left( 4 \right)\) in the graph is the maximum value in the smallest neighborhood of \(f\left( 4 \right)\) because \(f\left( 4 \right) = 5 \ge f\left( x \right)\) for all \(x\) nearest to \(4\).

By the definition of local maximum, \(f\left( 4 \right) = 5\) is the local maximum.

The point \(f\left( 6 \right)\) in the graph is the maximum value in the smallest neighborhood of \(f\left( 6 \right)\) because \(f\left( 6 \right) = 4 \ge f\left( x \right)\) for all \(x\) nearest to \(6\).

By the definition of local maximum, \(f\left( 6 \right) = 4\) is the local maximum.

Thus, the graph has no absolute minimum, the absolute maximum is at \(f\left( 4 \right) = 5\), the local maximum is at \(f\left( 4 \right) = 5\), and \(f\left( 6 \right) = 4\), and the local minimum is at \(f\left( 1 \right) = 3\), \(f\left( 2 \right) = 2\) and \(f\left( 5 \right) = 3\).