In Problems 49-56, for each graph of a function y = f ( x ) , find the absolute maximum and the absolute minimum, if they exist. Identify any local maximum values or local minimum values. 54.
In Problems 49-56, for each graph of a function y = f ( x ) , find the absolute maximum and the absolute minimum, if they exist. Identify any local maximum values or local minimum values. 54.
Solution Summary: The author explains how the graph can be used to find the absolute maximum and minimum of the given function.
In Problems 49-56, for each graph of a function
, find the absolute maximum and the absolute minimum, if they exist. Identify any local maximum values or local minimum values.
54.
Formula Formula A function f(x) attains a local maximum at x=a , if there exists a neighborhood (a−δ,a+δ) of a such that, f(x)<f(a), ∀ x∈(a−δ,a+δ),x≠a f(x)−f(a)<0, ∀ x∈(a−δ,a+δ),x≠a In such case, f(a) attains a local maximum value f(x) at x=a .
Expert Solution & Answer
To determine
To find: The following values using the given graph:
a. Absolute maximum and minimum if they exist.
b. Local maximum and minimum values.
Answer to Problem 50AYU
From the graph, the following results can be derived:
a. Absolute maximum is 4 but there is no absolute minimum.
b. Local maxima of the function is at and the value , and the local minimum point at and the corresponding values is .
Explanation of Solution
Given:
It is asked to find the absolute maximum and minimum of the given function and also identify its local maximum and minimum values.
Graph:
Interpretation:
a. Absolute maximum: The absolute maximum can be found by selecting the largest value of from the following list:
The values of at any local maxima of
in .
The value of at each endpoint of -that is, and .
It can be directly concluded from the graph and the definition that the curve has local maximum points and the corresponding values are .
The values of the local maximum at is 4.
The value of at each endpoint of that is, and the other end point is at minus infinity.
Therefore, the absolute maximum is same as the local maximum.
The absolute maximum of the function is 4.
Absolute minimum: The absolute minimum can be found by selecting the smallest value of from the following list:
The values of at any local minima of
in .
The value of at each endpoint of -that is, and .
It can be directly concluded from the graph and the definition that the curve has local minimum point at and the corresponding values is .
The value of at each endpoint of that is, and the other end point is at minus infinity.
Therefore, there is no absolute minimum for the function .
b. From the absolute maximum and absolute minimum values, identify the local extrema that the values of the local maximum at is and the local minimum point at and corresponding value is .
Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (4th Edition)
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