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 local maximum of ( y = f(x) ) is ( f(2)=6 ).
(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) ).
select the correct answer and, if necessary, fill in the answer boxes to complete your choice.
a. the local minimum of ( y = f(x) ) is ( f(square)=square ).
(type integers or simplified fractions.)

Explanation:

Step1: Recall the definition of local maximum

A local maximum of a function \(y = f(x)\) is a point where the function value is greater than or equal to the values of the function in a neighborhood around that point.
Looking at the graph, we can see that at \(x = 2\), the function \(y=f(x)\) has a local maximum. The \(y\) - value (function value) at \(x = 2\) is \(y = 6\). So \(f(2)=6\).

Step2: Check for other local maxima

We scan the graph for other points where the function "peaks" (i.e., has a higher value than its immediate neighbors). There is no other such point.

Step3: Recall the definition of local minimum

A local minimum of a function \(y = f(x)\) is a point where the function value is less than or equal to the values of the function in a neighborhood around that point.
Looking at the graph, we can see that at \(x=-2\) and \(x = 0\), the function \(y = f(x)\) has local minima. The \(y\) - value (function value) at \(x=-2\) is \(y = 2\) (so \(f(-2)=2\)) and at \(x = 0\) is \(y = 0\) (so \(f(0)=0\)).

Answer:

For the local maximum:
A. The local maximum of \(y = f(x)\) is \(f(2)=6\)
For the local minimum:
The local minima of \(y = f(x)\) are \(f(-2)=2\) and \(f(0)=0\)