1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Lerok [7]
3 years ago
15

Proof that x^y + y^x > 1 for all x,y > 0 ...?

Mathematics
1 answer:
bekas [8.4K]3 years ago
6 0
To prove this inequality we need to consider three cases. We need to see that the equation is symmetric and that switching the variables x and y does not change the equation.

Case 1: x >= 1, y >= 1

It is obvious that

 x^y >= 1, y^x >= 1
 x^y + y^x >= 2 > 1
 x^y + y^x > 1

Case 2: x >= 1, 0 < y < 1

 Considering the following sub-cases:

  - x = 1, x^y = 1
  - x > 1,

    Let x = 1 + n, where n > 0

    x^y = (1 + n)^y = f_n(y)

    By Taylor Expansion of f_e(y) around y = 0,

    x^y = f_n(0) + f_n'(0)/1!*y + f_n''(0)/2!*y^2 + ...
        = 1 + ln(1 + n)/1!*y + ln(1 + n)^2/2!*y^2 + ...

    Since ln(1 + n) > 0,

    x^y > 1

  Thus, we can say that x^y >= 1, and since y^x > 0.

  x^y + y^x > 1

  By symmetry, 0 < x < 1, y >= 1, also yields the same.

Case 3: 0 < x, y < 1

  We can prove this case by fixing one variable at a time and by invoking symmetry to prove the relation.

  Fixing the variable y, we can set the expression as a function,

  f(x) = x^y + y^x
  f'(x) = y*x^(y-1) + y^x*ln y 
  For all x > 0 and y > 0, it is obvious that
  f'(x) > 0.

  Hence, the function f(x) is increasing and hence the function f(x) would be at its minimum when x -> 0+ (this means close to zero but always greater than zero). 

  lim x->0+ f(x) = 0^y + y^0 = 0 + 1 = 1

  Thus, this tells us that 

  f(x) > 1.

  Fixing variable y, by symmetry also yields the same result: f(x) > 1.

  Hence, when x and y are varying, f(x) > 1 must also hold true.
    Thus, x^y + y^x > 1.

We have exhausted all the possible cases and shown that the relation holds true for all cases. Therefore, 

<span>  x^y + y^x > 1


----------------------------------------------------

I have to give credit to my colleague, Mikhael Glen Lataza for the wonderful solution.


I hope it has come to your help.

</span>
You might be interested in
Дешовок в гавне собак гандонов свиней)))))))))) xddddddd
Readme [11.4K]

Step-by-step explanation:

jhggcxddddddddddddddddd

8 0
2 years ago
Read 2 more answers
Jack's mother gave him 50 chocolates to give to his friends at his birthday party. He gave 3 chocolates to each of his friends a
djverab [1.8K]

50 = 3x + 2

50 - 2 = 3x

48 = 3x

48/3 = 3x/3

16 = x


Jack has 16 friends.

4 0
2 years ago
Read 2 more answers
Which equation is equivalent to
Anton [14]

Answer:

m { }^{2}  + 3m + 2 = 0

Step-by-step explanation:

ну где-то тааакккккккккк

8 0
3 years ago
Write a multiplication expression that can be used to solve 2/3a = 30.
Tresset [83]
<h3>Answer:</h3>

a=45

<h3>Solution:</h3>
  • We need to multiply both sides by 3 in order to get rid of the denominator, 3:
  • 2/3a•3=30•3
  • 2a=90
  • Divide both sides by 2:
  • a=45

Hope it helps.

Do comment if you have any query.

8 0
2 years ago
Which equation can be used to solve for x in the following diagram?
Olenka [21]

The equation we can use to find the value of x in the diagram given is: B. (5x + 30) + 5x = 90.

<h3>What are Complementary Angles?</h3>

Angles that give a sum of 90 degrees when added are referred to as complementary angles.

The angles, (5x + 30) and 5x are complementary angles, since the sum of both gives a right angle (90 degrees).

Therefore, the equation that we can use to find x in the diagram is: B. (5x + 30) + 5x = 90.

Learn more about complementary angles on:

brainly.com/question/16281260

#SPJ1

8 0
2 years ago
Other questions:
  • How do I find the area of a trapezoid?
    10·2 answers
  • 26+6b greater than or equal to 2(3b+4)
    11·1 answer
  • um time de futsal marcou 60 gols, correspondendo a quatro sextos do total de gols do campeonato. Quantos gols foram marcados no
    13·1 answer
  • Two trains leave stations 324 miles apart at the same time and travel toward each other. one train travels at 95 mph while the o
    7·1 answer
  • The polynomial (x - 2) is a factor of the polynomial 4x2 - 6x-4. True or false?
    10·1 answer
  • Is 0.35 l greater than 300 ml
    15·1 answer
  • Assume that y varies inversely<br> with x. If y= 16 when x = 0.5,<br> find y when x = 32.<br> y=[? ]
    14·1 answer
  • Please help out with this Geometry problem !!!
    14·2 answers
  • Which of the following is a fully factored expression of 24x – 56y + 72?
    14·1 answer
  • How do you find the HCF of 15t and 6s?
    15·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!