Question

(a) Sketch the graph of a function on \(\left( { - 1,2} \right)\) that has an absolute maximum but no local maximum.

(b) Sketch the graph of a function on \(\left( { - 1,2} \right)\) that has a local maximum but no absolute maximum.

Short answer

(a) The graph of a function \(f\)on \(\left( { - 1,2} \right)\)that has an absolute maximum, but no local maximum is shown below:

(b) The graph of a function \(f\) on \(\left( { - 1,2} \right)\) that has a local maximum, but no absolute maximum is shown below:

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 minimumvalue 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 minimum value 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 maximumvalue of \(f\) if \(f\left( c \right) \ge f\left( x \right)\) when \(x\) is near \(c\).

Describe the given properties to graph the function

Given that the function has an absolute maximum at 2 and no local maximum.

The given interval is a closed interval.

ByThe Extreme value theorem,for a continuous function \(f\)on a closed interval \(\left( {a,b} \right)\), the function \(f\) attains the absolute minimum and maximum values.

Sketch the graph of the function

The graph of a function \(f\)is shown below:

(b) Step 5: Describe the given properties to graph the function

Given that the function has a local maximum and no absolute maximum.

The graph of \(f\)has local maximum, so, the graph of \(f\)changes from increases to decreases.

ByThe Extreme value theorem,for a continuous function\(f\)on a closed interval \(\left( {a,b} \right)\), the function \(f\) attains the absolute minimum and maximum values.

The function has no absolute maximum means the graph of the function has breakpoints.

Sketch the graph of the function