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. 52.
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. 52.
Solution Summary: The author explains how the graph can be used to find the absolute maximum and minimum of a given function and identify its local minimum and maximum values.
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.
52.
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 48AYU
From the graph, the following results can be derived:
a. There is no absolute maximum point but the absolute minimum is 1.
b. There is no local maximum and local minimum point.
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 . But this point is disconnected.
The values of the local maximum at is 4. Therefore, the local maximum point is disconnected point.
The value of at each endpoint of that is, .
The largest of these, 4, but the point is disconnected. 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
The value of at each endpoint of that is, .
The smallest is, 1, is the absolute minimum.
b. From the absolute maximum and absolute minimum values, identify the local extrema that there is no local maximum value and local minimum values.
Single Variable Calculus: Early Transcendentals (2nd Edition) - Standalone book
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.