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. 53.
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. 53.
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.
53.
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 49AYU
From the graph, the following results can be derived:
a. There is no absolute maximum point but the absolute minimum is 0.
b. Local maxima of the function is at and the value , and the local minimum point at and and the corresponding values are
and .
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 at and the corresponding values are .
The values of the local maximum at is 3.
The value of at each endpoint of that is, and the other end point is at infinity.
Therefore, there is no absolute maximum point for the function .
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 and the corresponding values are
and .
The value of at each endpoint of that is, and the other end point is at infinity.
The smallest is, 1, is the absolute minimum.
b. From the absolute maximum and absolute minimum values, identify the local extrema that the values of the local minimum at is and the local minimum point at and and the corresponding values are and .
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