Does anyone happen to know a nice way to show that (a+b)^n<=a^n+b^n, where a,b>=0 and n in (0,1]? I figured integrating might help, but I've been unable to pull my argument full circle. Any suggestions are appreciated :)

Almintas2l 2022-07-17 Answered
How to show ( a + b ) n a n + b n , where a , b 0 and n ( 0 , 1 ]?
Does anyone happen to know a nice way to show that ( a + b ) n a n + b n , where a , b 0 and n ( 0 , 1 ]? I figured integrating might help, but I've been unable to pull my argument full circle. Any suggestions are appreciated :)
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (2)

slapadabassyc
Answered 2022-07-18 Author has 21 answers
We have
1 = a a + b + b a + b ( a a + b ) n + ( b a + b ) n
for 0 < n 1.

We have step-by-step solutions for your answer!

Avery Stewart
Answered 2022-07-19 Author has 2 answers
The cases a = 0 is trivial. For a 0 let m = 1 / n and c = a 1 / m = a n and d = b 1 / m = b n . Then
( a + b ) n a n + b n ( c m + d m ) 1 / m c + d
c m + d m ( c + d ) m 1 + ( d / c ) m ( 1 + d / c ) m
( d / c ) m ( 1 + d / c ) m 1
m 0 d / c x m 1 d x m 0 d / c ( 1 + x ) m 1 d x
which holds as d / c 0 and m 1 0.

We have step-by-step solutions for your answer!

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-05-26
If a b = b c = c d , prove that a d = a 5 + b 2 c 2 + a 3 c 2 b 4 c + d 4 + b 2 c d 2
What I've done so far;
a b = b c = c d = k a = b k , b = c k , c = d k a = c k 2 , b = d k 2 a = d k 3
I tried substituting above values in the right hand side of the equation to get a d but couldn't.
asked 2022-07-07
How to find the value of a and b from this limit problem with or without L'Hopital's formula?
Consider the limit:
lim x 4 x 2 + a x + b x 4 = 14
Question: How can I find the values of a and b?
Attempt:
My first thought is, we need to use L'Hopital's rule to make sure that the denominator isn't zero:
Applying L'Hopital's rule, and we get:
lim x 4 2 x + a 1 = 14
Then, we can substitute the limit of x to the equation such that:
2 ( 4 ) + a = 8 + a = 14
and we get that the value of a is 6.
But, how can I find the value of b? It seems that after applying the L'Hopital's formula the value of b disappears.
Also, is there a way to solve this problem without L'Hopital's rule?
Thanks
asked 2022-06-25
Algebraic Problem Fractions
Given the following fraction,
n 2 n 1 ( 2 ) n 1
Is it correct that it simplifies to the following fraction,
( 1 ) n 1 n
I am in doubt because I have checked on three different places and they all gave different answer.
asked 2021-08-11

For each of the sets below, determine whether {2} is an element of that set.

a) {xR x is an integer greater than 1}
b) {xR  x is the square of an integer}

c) {2,{2}}

d) {{2},{{2}}}

e) {{2},{2,{2}}}

f) {{{2}}}

asked 2022-07-04
Can fractions be written with a one under them and can problems be solved this way
How would you write a 0.1 ?
Are either of these ways acceptable?
a 1 1 10 1 = a 1 × 1 1 10
or
a 1 1 10 = a 1 × 10 1
asked 2022-05-13
What must be added to this mixed fraction to produce this mixed fraction
This may seem like a simple question which is fit for a Google search; however, I'm unable to find an answer.
What must be added to 6 3 5 to produce 3 5 6
I hope I'll finally understand. Step-by-step solution is much appreciated.
asked 2021-11-16
Simplify the complex fraction.
207x1021x=?