Question

(3.3 - 3.4)
8 of 10
this quiz: 10 point(s) possible
this question: 1 point(s) possible
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.
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.
a. the local maximum of ( y = f(x) ) is ( f(square)=square ).
(type integers or simplified fractions.)
b. there are two local maxima. the leftmost maximum is ( f(square)=square ) the rightmost maximum is and ( f(square)=square )
(type integers or simplified fractions.)
c. there is no local maximum for ( y = f(x) ).

Explanation:

Step1: Understand local maximum

A local maximum is a point where the function changes from increasing to decreasing.

Step2: Analyze the graph

Looking at the graph, there are two points where the function changes from increasing to decreasing. The left - most one is at \(x = 2\) with \(y=f(2)=6\) and the right - most one (if we consider the end - behavior, but since the right - hand side is going up without bound in the given view, we focus on the non - end - point local maxima). Wait, no, actually, in the given graph, there are two local maxima. The left - most local maximum is at the point \((2,6)\) (so \(f(2) = 6\)) and the right - hand side, if we assume the graph is as shown (the other point is not a local maximum in the traditional sense as the function is increasing after \(x = 4\) towards the upper - right in the given graph view. Wait, no, re - checking: A local maximum is a value \(f(c)\) such that \(f(c)\geq f(x)\) for all \(x\) in some open interval containing \(c\). In the given graph, at \(x=-2\), the function has a value \(y = 2\), but it is a local minimum. At \(x = 2\), \(y=6\) (a local maximum) and as the function is increasing after \(x = 4\) (towards the upper - right in the graph), there is no other local maximum. Wait, no, actually, if we consider the definition of local maximum: a point \((a,f(a))\) is a local maximum if there exists an open interval \(I\) containing \(a\) such that \(f(a)\geq f(x)\) for all \(x\in I\). In the given graph, the point \((2,6)\) is a local maximum.

Step1: Identify the lowest \(y\) - value

We compare the \(y\) - values of the key points \((-2,2)\), \((0,0)\), \((4,4)\).
Since \(0<2\) and \(0 < 4\), and \(f(0)\) is calculated as \(y\) - value when \(x = 0\) (from the point \((0,0)\))

Answer:

B. There are two local maxima. The leftmost maximum is \(f(2)=6\) the rightmost maximum is (there is no rightmost maximum as the function is increasing towards the upper - right in the given graph view, but if we assume the problem's graph (maybe a mis - interpretation), no, wait, re - checking the graph: The point \((- 2,2)\) is a local minimum (function changes from decreasing to increasing), \((2,6)\) is a local maximum (function changes from increasing to decreasing), \((4,4)\) is a local minimum (function changes from decreasing to increasing). So the local maxima: only \(f(2)=6\). But looking at the options:
If we assume the first part (absolute minimum):
For the absolute minimum:

Step1: Understand absolute minimum

The absolute minimum of a function \(y = f(x)\) on its domain is the smallest value of \(y\).

Step2: Analyze the graph

Looking at the points \((-2,2)\), \((0,0)\), \((4,4)\). The value \(y = 0\) at \(x = 0\) (since \(f(0)=0\) and \(0<2\) and \(0 < 4\)).
For the local maximum:
The local maximum occurs at \(x = 2\) where \(f(2)=6\)

Absolute minimum:
A. The absolute minimum of \(y = f(x)\) is \(f(0)=0\)
Local maximum:
A. The local maximum of \(y = f(x)\) is \(f(2)=6\)

So, for the absolute minimum part: