Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Suppose f:R-IR is a diffeetiable function which satısfies flo) = 0 We wart to show f': IR → IR is a coustant fanction if and cnly if fl*+yl =fx9 + fiy) for all ×, y € IR

Suppose f:R-IR is a diffeetiable function which satısfies flo) = 0 We wart to show f': IR → IR is a coustant fanction if and cnly if fl*+yl =fx9 + fiy) for all ×, y € IR

Intermediate Algebra
Intermediate Algebra
10th Edition
ISBN:9781285195728
Author:Jerome E. Kaufmann, Karen L. Schwitters
Publisher:Jerome E. Kaufmann, Karen L. Schwitters
Chapter9: Functions
Section9.1: Relations And Functions
Problem 76PS
See similar textbooks
icon
Related questions
Question
100%

Please show the detailed steps. I would rate asap.

Suppose f.R→IR is a diffetinble funtion which satısfies flo) = 0 We want to show f': IR → IR is a coustant function if and anly if flx+yl =frx + fiy) for all ×, y & IR
Transcribed Image Text:Suppose f.R→IR is a diffetinble funtion which satısfies flo) = 0 We want to show f': IR → IR is a coustant function if and anly if flx+yl =frx + fiy) for all ×, y & IR
Expert Solution
Step 1

Given that f:ℝ→ℝ is a differentiable function which satisfies f0=0.

It is required to show that f' : ℝ→ℝ is constant function if and only if fx+y=fx+fy for all x, y∈ℝ.

Suppose if f' : ℝ→ℝ is constant function.

Then it is required to prove that  fx+y=fx+fy for all x, y∈ℝ.

Since f' : ℝ→ℝ is constant function, f'x=a for all x∈ℝ.

Integrating on both sides with respect to x will give

f(x)=ax+b         ...... (1)

where b is the integration constant.

Given that the function satisfies f0=0.

Substituting x=0 in (1) will give

⇒f(0)=a0+b⇒     0=0+b⇒     0=b

Thus the integration constant b=0.

Then (1) becomes f(x)=ax

Now for all x, y∈ℝ,

f(x+y)=ax+y

            =ax+ay

            =fx+fy

Hence proved that if  f' : ℝ→ℝ is constant function, then fx+y=fx+fy for all x, y∈ℝ.        .......... (A)

steps

Step by step

Solved in 2 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Knowledge Booster
Basics (types, similarity, etc)
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Intermediate Algebra
Intermediate Algebra
Algebra
ISBN:
9781285195728
Author:
Jerome E. Kaufmann, Karen L. Schwitters
Publisher:
Cengage Learning
Algebra for College Students
Algebra for College Students
Algebra
ISBN:
9781285195780
Author:
Jerome E. Kaufmann, Karen L. Schwitters
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
College Algebra
College Algebra
Algebra
ISBN:
9781938168383
Author:
Jay Abramson
Publisher:
OpenStax
Elements Of Modern Algebra
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
College Algebra
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS