In Problem 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. 51.
In Problem 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. 51.
In Problem 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.
51.
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 47AYU
From the graph, the following results can be derived:
a. The absolute maximum is 4 and the absolute minimum is 1.
b. Local maxima of the function is at and the value , also the local minima of the function is at and the value .
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 point at .
The values of the local maximum at is 4. Therefore, the local maximum point is .
The value of at each endpoint of and -that is, and .
The largest of these, 4, is the absolute maximum.
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 .
The values of the local minimum at is 1. Therefore, the local minimum point is .
The value of at each endpoint of and -that is, and .
The largest of these, 1, is the absolute minimum.
b. From the absolute maximum and absolute minimum values, identify the local extrema that is the local maxima point is at , the value is and the local minima point is at , the value is .
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