1. A strictly monotonic function is defined as a function that either always increases or always decreases in its domain. Suppose we consider all strictly monotonic functions that have (-0, ) as their domain. Explain intuitively why it is true that for all such functions, there is no point in the domain where the tangent line is horizontal. Now recall a specific class of functions called one-one functions. These are functions that have exactly one input corresponding to every output. Explain intuitively why the exact same conclusion about tangent lines can be made for one-one functions as well. Is this enough to say that a strictly monotonic function is always one-one and vice- versa? Discuss.

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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1. A strictly monotonic function is defined as a function that either always increases or
always decreases in its domain. Suppose we consider all strictly monotonic functions
that have (-, 0) as their domain. Explain intuitively why it is true that for all such
functions, there is no point in the domain where the tangent line is horizontal. Now
recall a specific class of functions called one-one functions. These are functions that
have exactly one input corresponding to every output. Explain intuitively why the
exact same conclusion about tangent lines can be made for one-one functions as well.
Is this enough to say that a strictly monotonic function is always one-one and vice-
versa? Discuss.
Transcribed Image Text:1. A strictly monotonic function is defined as a function that either always increases or always decreases in its domain. Suppose we consider all strictly monotonic functions that have (-, 0) as their domain. Explain intuitively why it is true that for all such functions, there is no point in the domain where the tangent line is horizontal. Now recall a specific class of functions called one-one functions. These are functions that have exactly one input corresponding to every output. Explain intuitively why the exact same conclusion about tangent lines can be made for one-one functions as well. Is this enough to say that a strictly monotonic function is always one-one and vice- versa? Discuss.
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