Question

use the graph of a function ( y = f(x) ) to find the absolute maximum and the absolute minimum, if they exist. identify any local maximum values or local minimum values.
select the correct answer and, if necessary, fill in the answer boxes to complete your choice.
a. the absolute maximum of ( y = f(x) ) is ( f(square)=square ). (type integers or simplified fractions.)
b. there is no absolute maximum for ( y = f(x) ).
select the correct answer and, if necessary, fill in the answer boxes to complete your choice.
a. the absolute minimum of ( y = f(x) ) is ( f(square)=square ). (type integers or simplified fractions.)
b. there is no absolute minimum for ( y = f(x) ).
select the correct answer and, if necessary, fill in the answer boxes to complete your choice.

Explanation:

Step1: Analyze the absolute maximum

The absolute maximum of a function is the largest \(y -\)value of the function over its entire domain. Looking at the graph, as \(x\) approaches \(6\) (the right - hand end of the domain shown), the function value increases without bound. So, there is no absolute maximum.

Step2: Analyze the absolute minimum

The absolute minimum of a function is the smallest \(y -\)value of the function over its entire domain. From the points \((-2,2)\), \((0,1)\), \((2,6)\), \((3,4)\), the \(y -\)value at \(x = 0\) is \(y=1\). So, \(f(0)=1\) is the absolute minimum.

Step3: Analyze the local maximum

A local maximum is a point where the function changes from increasing to decreasing. At \(x = 2\), the function changes from increasing (for \(x<2\)) to decreasing (for \(x\) between \(2\) and \(3\)). So, \(f(2)=6\) is a local maximum.

Step4: Analyze the local minimum

A local minimum is a point where the function changes from decreasing to increasing. At \(x = 0\), the function changes from decreasing (for \(x < 0\)) to increasing (for \(x>0\)). So, \(f(0)=1\) is a local minimum. Also, at \(x = 3\), the function changes from decreasing (for \(x\) between \(2\) and \(3\)) to increasing (for \(x>3\)), so \(f(3)=4\) is a local minimum.

Answer:

For the absolute maximum: B. There is no absolute maximum for \(y = f(x)\).
For the absolute minimum: A. The absolute minimum of \(y = f(x)\) is \(f(0)=1\).
For the local maximum: A. The local maximum value of \(y = f(x)\) is \(f(2)=6\).
For the local minimum: A. The local minimum values of \(y = f(x)\) are \(f(0)=1\) and \(f(3)=4\).