Proving a/b+b/a >= 2 using Mathematical Proof | Homework Help

In summary, to prove that a/b+b/a is greater than or equal to 2, you can use the formula (a^2+b^2)/ab and simplify it to (a-b)^2>=0. This can be further simplified to (a-b)^2=0 when a=b and (a-b)^2>0 when a does not equal b. Therefore, (a-b)^2>=0 for all values of a and b, proving that a/b+b/a is greater than or equal to 2.
  • #1
TheMathNoob
189
4

Homework Statement


Let a,b be in the positive reals. Prove a/b+b/a is >=2

Homework Equations

The Attempt at a Solution


I have no idea. Maybe add the two ratios: (a^2+b^2)/a*b and then try to analyze separately the numerator and denominator?
 
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  • #2
Those a's and b's in the denominator are pesky. Why not multiply through by ab and see what that gives you?
 
  • #3
jbriggs444 said:
Those a's and b's in the denominator are pesky. Why not multiply through by ab and see what that gives you?
I feel like this is related to the law of cosines
 
  • #4
TheMathNoob said:
I feel like this is related to the law of cosines
Your powers of pattern recognition are good, but there is another formula involving a2, b2 and 2ab that is simpler yet.
 
  • #5
jbriggs444 said:
Your powers of pattern recognition are good, but there is another formula involving a2, b2 and 2ab that is simpler yet.
oh hahahahahah (a-b)^2>=0
when a=b (a-b)^2=0
when a!=b (a-b)^2>0
so (a-b)^2>=0
 

1. What is the purpose of proving a/b+b/a >= 2 using mathematical proof?

The purpose of proving a/b+b/a >= 2 using mathematical proof is to demonstrate the validity of this mathematical statement and show that it holds true for all possible values of a and b. This proof also helps to understand the underlying mathematical concepts and principles involved in the expression.

2. How do you approach the proof of a/b+b/a >= 2 using mathematical proof?

The proof of a/b+b/a >= 2 can be approached using various mathematical techniques such as algebraic manipulation, mathematical induction, or contradiction. It is essential to understand the properties of fractions and basic arithmetic rules to successfully prove this statement.

3. Can you explain the steps involved in proving a/b+b/a >= 2 using mathematical proof?

To prove a/b+b/a >= 2 using mathematical proof, we start by assuming that a and b are positive real numbers. Then, we can manipulate the expression using algebraic operations to transform it into a simpler form that is easier to prove. Finally, we use mathematical principles and properties to show that the inequality holds true for all possible values of a and b.

4. What are some common mistakes to avoid when proving a/b+b/a >= 2 using mathematical proof?

Some common mistakes to avoid when proving a/b+b/a >= 2 using mathematical proof include assuming that the statement is true without proof, using incorrect algebraic manipulations, or making assumptions about the values of a and b. It is important to carefully follow each step of the proof and justify each step using mathematical principles.

5. Can the proof of a/b+b/a >= 2 be extended to other similar mathematical statements?

Yes, the proof of a/b+b/a >= 2 can be extended to other similar mathematical statements involving fractions and inequalities. By understanding the underlying principles and techniques used in this proof, we can apply them to solve and prove other mathematical statements with similar structures.

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