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(quad)=quad ).
(type integers or simplified fractions.)
b. the local maxima of ( y = f(x) ) are ( f(quad)=quad ) and ( f(quad)=quad ).
(type integers or simplified fractions.)
c. there is no local maximum for ( y = f(x) ).

Explanation:

Step1: Understand 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.

Step2: Analyze the graph

Looking at the graph of \(y = f(x)\), we have two points that satisfy the local - maximum condition. At \(x = 1\), \(y=f(1)=6\) and in a small neighborhood around \(x = 1\), \(f(1)\) is the largest value. At \(x = 3\), \(y = f(3)=4\) and in a small neighborhood around \(x = 3\), \(f(3)\) is the largest value.

Answer:

B. The local maxima of \(y = f(x)\) are \(f(1)=6\) and \(f(3)=4\)