Real Analysis 1- Using the 2nd photo solution as an example, solve similarly the problem 6B 6. Choose and work either (a) or (b), NOT BOTH.(a) Let f : R? → R? be defined by f(x, y) = (x+ y,x – y). Use the definition toprove that f is continuous at (1, –1).(b) Use the Mean Value Theorem on the interval [0, x] for x > 0, to show thatIn(1+x) < x. -6-4. For x+0, lhunsmcax- sinbx< la-blPao: let ito ke faxed and lat f: [a, b]R bedefnisedby fe0= eintx).then frlCom tinunyLa,6], differeukelle enl,b)Laib),on la,6)a theMean Value Thermeum,f(6)-f(6)-ske) whereb-aprestsmce f'ct)=x caftx)fl6)-fca)4.bnaThusSinex)-Sincax)6-aSamlex)- SmaxLktb9orSuiCaxt- Sm (bx)2 lb-al-la -b

The given problem is to show the inequality ln(1+x)≤x using the mean value theorem in the interval…

Answered: (b) Use the Mean Value Theorem on the…

Math

Advanced Math

(b) Use the Mean Value Theorem on the interval [0, x] for. x > 0, to show that In(1+x) < x.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.5: The Kernel And Range Of A Linear Transformation

Problem 30EQ

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Question

Real Analysis 1- Using the 2nd photo solution as an example, solve similarly the problem 6B

6. Choose and work either (a) or (b), NOT BOTH.
(a) Let f : R? → R? be defined by f(x, y) = (x+ y,x – y). Use the definition to
prove that f is continuous at (1, –1).
(b) Use the Mean Value Theorem on the interval [0, x] for x > 0, to show that
In(1+x) < x.

-6-
4. For x+0, lhun
smcax- sinbx
< la-bl
Pao: let ito ke faxed and lat f: [a, b]R be
defnised
by fe0= eintx).
then frl
Com tinuny
La,6], differeukelle enl,b)
Laib),
on la,6)
a the
Mean Value Thermeum,
f(6)-f(6)-ske) where
b-a
prest
smce f'ct)=x caftx)
fl6)-fca)
4.
bna
Thus
Sinex)-Sincax)
6-a
Samlex)
- SmaxLkt
b9
or
SuiCaxt- Sm (bx)
2 lb-al-la -b

Branch of mathematical analysis that studies real numbers, sequences, and series of real numbers and real functions. The concepts of real analysis underpin calculus and its application to it. It also includes limits, convergence, continuity, and measure theory.

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