Question

(a) Sketch the graph of a function that has two local maxima, one local minimum and no absolute minimum.

(b) Sketch the graph of a function that has three local minima, two local maxima, and seven critical numbers.

Short answer

(a) The graph of a function \(f\)that has two local maxima, one local minimum, and no absolute minimum is shown below:

(b) The graph of a function \(f\) that has three local minima, two local maxima, and seven critical numbers 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 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 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.

The function has no absolute minimum means the graph of the function has break points.

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 three local minima, two local maxima, and seven critical numbers.

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

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