Question

For each of the numbers \(a,\,b,\,c,\,d,\,r\) and \(s\) state whether the function whose graph is shown has an absolute maximum or minimum, a local maximum or minimum, or neither a maximum nor a minimum.

Short answer

The absolute minimum is at \(r\), absolute maximum is at \(s\), the local maximum is at \(c\) , and the local minimum is at \(b\), and \(r\). The graph has neither a maximum nor minimum is at \(a\), and \(d\).

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 theabsolute 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 thelocal minimum value of \(f\) if \(f\left( c \right) \le f\left( x \right)\) when \(x\) is near \(c\).

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

Find the absolute minimum and maximum value

Consider a graph of the function \(f\left( x \right)\).

The point \(f\left( r \right)\) in the graph is the absolute minimum in \(f\left( D \right)\) where \(D\) is the domain because \(f\left( r \right) \le f\left( x \right)\) for all \(x\) in \(D\).

The point \(f\left( s \right)\) in the graph is the absolute maximum in \(f\left( D \right)\) where \(D\) is the domain because \(f\left( s \right) \ge f\left( x \right)\) for all \(x\) in \(D\).

Find the local minimum and maximum value

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

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

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

Find the points that is neither absolute minimum and maximum nor local minimum and maximum

The point \(f\left( a \right)\) in the graph is not an absolute minimum because \(f\left( r \right)\) is the absolute minimum.

The point \(f\left( a \right)\) in the graph is not an absolute maximum because \(f\left( s \right)\) is the absolute maximum.

The point \(f\left( a \right)\) in the graph is not a local minimum because \(f\left( a \right)\) does not defined to the left side of the nearest point of \(a\).

The point \(f\left( a \right)\) in the graph is not a local maximum because \(f\left( a \right)\) does not define to the left side of the nearest point of \(a\).

The point \(f\left( d \right)\) in the graph is not an absolute minimum because \(f\left( r \right)\) is the absolute minimum.

The point \(f\left( d \right)\) in the graph is not an absolute maximum because \(f\left( s \right)\) is the absolute maximum.

The point \(f\left( d \right)\) in the graph is not a local minimum because \(f\left( d \right) \le f\left( x \right)\) to the left side at the nearest point of \(d\) and \(f\left( d \right) \ge f\left( x \right)\) to the right side at the nearest point of \(d\).

The point \(f\left( d \right)\) in the graph is not a local maximum because \(f\left( d \right) \le f\left( x \right)\) to the left side at the nearest point of \(d\) and \(f\left( d \right) \ge f\left( x \right)\) to the right side at the nearest point of \(d\).

Thus, the absolute minimum is at \(r\), absolute maximum is at \(s\), the local maximum is at \(c\) , and the local minimum is at \(b\), and \(r\). The graph has neither a maximum nor minimum is at \(a\), and \(d\).