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.
a. the local maximum of ( y = f(x) ) is ( f(2)=4 ). (type integers or simplified fractions.)
b. there are two local maxima. the leftmost maximum is ( f()= ) the rightmost maximum is and ( f()= ). (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()= ) (type integers or simplified fractions.)
b. there are two local minima. the leftmost minimum is ( f()= ) and the rightmost minimum is ( f()= ) (type integers or simplified fractions.)
c. there is no local minimum for ( y = f(x) ).

Explanation:

Step1: Analyze local maxima

A local maximum is a point where the function changes from increasing to decreasing. From the graph, at \(x = 2\), \(y=f(2)=4\) is a local maximum. There is no other point where the function has a "peak" (changes from increasing to decreasing) in the given visible part of the graph (excluding the far - right end which goes to infinity). So, option A for local maxima is correct.

Step2: Analyze local minima

A local minimum is a point where the function changes from decreasing to increasing. The function has two such points: at \(x=-1\), \(y = f(-1)=2\) (left - most local minimum) and at \(x = 0\), \(y=f(0)=1\) (right - most local minimum in the non - extended part of the graph we can analyze for minima).

Answer:

For local maxima: A. The local maximum of \(y = f(x)\) is \(f(2)=4\).
For local minima: B. There are two local minima. The leftmost minimum is \(f(-1)=2\) and the rightmost minimum is \(f(0)=1\).