How do you prove the inequality (a^3+b^3)(a^2-ab+b^2) <= a^5+b^5?

In summary, the problem states that for all positive values of a and b, if a^3+b^3 = a^5+b^5, then it can be proven that a^2+b^2<=1+ab using various methods such as trigonometry or creative long division. One method involves showing that (a-b)^2(b+a) is non-negative, which leads to the desired result.
  • #1
siddharthmishra19
27
0
The problem

For all a>0 and b>0 if a^3+b^3 = a^5+b^5 prove that
a^2+b^2<=1+ab

I have no idea on even how to start... i have tried using trigonometry (like in previous post) but come to a dead end... i am looking for the simplest method...
 
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  • #2
The only proof I've been able to come up with is kind of weird. Maybe you can do better. You should easily be able to convince yourself that if both a and b are >1 the equality can't hold. Same for <1. So we can take a>=1 and b<=1. Now consider f(n)=a^n+b^n (n>0). Now show f(n) has one extermum and it's a minimum. Your premise says f(3)=f(5). This says the extremal n is between 3 and 5. So f(1)>=f(3). Or a+b>=a^3+b^3. Factor the LHS and divide by (a+b) and you have your result. Funny, huh?
 
  • #3
I tried starting with a different approach. Since LHS = RHS, then RHS/LHS = 1. Doing a little creative long division, I obtained:

[tex]\frac{a^5+b^5}{a^3+b^3} = 1 = a^2 + b^2 - \frac{a^3 b^2 + a^2 b^3}{a^3 + b^3}[/tex]

From there,
[tex] 1 + \frac{a^3 b^2 + a^2 b^3}{a^3 + b^3} = a^2 + b^2 [/tex]

Then, perhaps on the LHS, factor out an ab:
[tex] 1 + ab( \frac{a^2 b + a b^2}{a^3 + b^3}) = a^2 + b^2 [/tex]

I'd think you can turn it around to [tex]a^2 + b^2 = 1 + ab( \frac{a^2 b + a b^2}{a^3 + b^3}) = (re-written) <= re-written with a term dropped out. [/tex]

I'm drawing a momentary blank (and have to get home!) but I can't "see" the next step from here; but maybe it's enough that you can continue.
 
Last edited:
  • #4
Keep trying, drpizza. I'd love to see the non-calc proof. I tried stuff like that for quite a while.
 
  • #5
It's equivalent to show that:

[tex](a^3 + b^3)(a^2 - ab + b^2) \leq a^5 + b^5[/tex]

This line is true iff:

[tex]a^5 + a^2b^3 - a^4b - ab^4 + a^3b^2 + b^5 \leq a^5 + b^5[/tex]

iff

[tex]a^2b^3 - a^4b - ab^4 + a^3b^2 \leq 0[/tex]

iff

[tex]ab^2 - a^3 - b^3 + a^2b \leq 0[/tex]

iff

[tex]a(b^2 - a^2) - b(b^2 - a^2) \leq 0[/tex]

iff

[tex](a-b)(b^2 - a^2) \leq 0[/tex]

iff

[tex](a-b)(b-a)(b+a) \leq 0[/tex]

iff

[tex]-(a-b)(a-b)(b+a) \leq 0[/tex]

iff

[tex](a-b)^2(b+a) \geq 0[/tex]

Well (a-b)2 is a square, hence non-negative. b and a are both positive, so (b+a) is positive. So the product on the left is indeed non-negative, so the desired result holds.
 

What is the concept of inequality in mathematics?

In mathematics, inequality refers to a relationship between two quantities where one is greater or less than the other. It is denoted by symbols such as <, >, ≤, and ≥.

How can inequality be proved in mathematics?

Inequality can be proved using various methods such as algebraic manipulation, mathematical induction, and logical reasoning. It involves showing that one side of the inequality is always greater or less than the other side.

What is the importance of proving inequalities?

Proving inequalities is important in mathematics as it helps in understanding the relationship between different quantities and their properties. It also allows for the comparison of values and helps in solving various mathematical problems.

What are some common techniques used in proving inequalities?

Some common techniques used in proving inequalities include the use of basic arithmetic operations, substitution, and the use of mathematical properties such as the triangle inequality and the Cauchy-Schwarz inequality.

Are there any real-life applications of proving inequalities?

Yes, there are many real-life applications of proving inequalities. For example, in economics, inequalities are used to study the distribution of wealth and income in a society. In physics, inequalities are used to determine the limits of physical phenomena. In engineering, they are used to design efficient systems with optimal performance.

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