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(a) Show that e>1+ x if x> 0. [Hint: Show that f(x) = e* – (1 + x) is increasing for x> 0.] (b) Deduce that s e**dx < e.

(a) Show that e>1+ x if x> 0. [Hint: Show that f(x) = e* – (1 + x) is increasing for x> 0.] (b) Deduce that s e**dx < e.

Calculus: Early Transcendentals
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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(a) Show that e>1+ x if x> 0. [Hint: Show that f(x) = e* – (1 + x) is increasing for x> 0.] (b) Deduce that s e**dx < e.
Transcribed Image Text:(a) Show that e>1+ x if x> 0. [Hint: Show that f(x) = e* – (1 + x) is increasing for x> 0.] (b) Deduce that s e**dx < e.
Expert Solution
Step 1

If x≥0 as exp is increase we have ex≥1. they by means value theorem there exist c∈0,x such that

                             ex-1=ecx

Thus

                            ex≥x+1

If x≤-1, we have

                           ex≥0>1+x

Now if -1<x<0 we have

                         ec<1⇒-ec>-1⇒-ecx<-x

Thus

                      1-ex=ec(-x)<-x

and

                    ex>1+x

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