Math Is Fun Forum

Stas B.

Guest

Some More Math Problems

This time I need to prove the following:
a*b + a*c + b*c <= a^2 + b^2 + c^2

Any ideas how that can be done?

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Some More Math Problems

Well you might imagine a, b, and c to be the sides of a cube with rectangular sides.
Then a*b, a*c, and b*c are the areas of three sides of the cube.
Next, I don't know what comes next...

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igloo myrtilles fourmis

luca-deltodesco

Member

Registered: 2006-05-05

Posts: 1,470

Re: Some More Math Problems

This time I need to prove the following:
a*b + a*c + b*c <= a^2 + b^2 + c^2

Any ideas how that can be done?

well..... lets see

given that a>0, b>0, c>0

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The Beginning Of All Things To End.
The End Of All Things To Come.

Stas B.

Guest

Re: Some More Math Problems

Wait, but what does that give us?

luca-deltodesco

Member

Registered: 2006-05-05

Posts: 1,470

Re: Some More Math Problems

Wait, but what does that give us?

nothing that was just me trying to get it

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The Beginning Of All Things To End.
The End Of All Things To Come.

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Some More Math Problems

I got it!!
Make a, b, and c these names:  M for middle #.  M + x  for large number.   M - y  for small number.
Now x and y are greater than or equal to zero.
Substitute in and expand and cancel stuff and you get
the result that the big side of the equation is bigger by x^2 + xy + y^2.
No time to explain.  But seems to be a positive amount bigger.

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igloo myrtilles fourmis

luca-deltodesco

Member

Registered: 2006-05-05

Posts: 1,470

Re: Some More Math Problems

This time I need to prove the following:
a*b + a*c + b*c <= a^2 + b^2 + c^2

Any ideas how that can be done?

I got it!!
Make a, b, and c these names:  M for middle #.  M + x  for large number.   M - y  for small number.
Now x and y are greater than or equal to zero.
Substitute in and expand and cancel stuff and you get
the result that the big side of the equation is bigger by x^2 + xy + y^2.
No time to explain.  But seems to be a positive amount bigger.

ok, by that

since x,y >= 0, -xy will ALWAYS be negative
and even if not, x^2 + y^2 will ALWAYS be positive  (unless there 0, in which case 0 is <= 0 anyways)

therefore

a*b + a*c + b*c <= a^2 + b^2 + c^2

goodjob frank

(rewritten with math tags)

Last edited by luca-deltodesco (2006-06-12 10:17:20)

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The Beginning Of All Things To End.
The End Of All Things To Come.

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: Some More Math Problems

Great job! Franklin!

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X'(y-Xβ)=0

Stas B.

Guest

Re: Some More Math Problems

Thanks Frank!
You're brilliant!

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Some More Math Problems

Thanks for the thank you's!  Just sorry I don't explain myself better, but luca did a nice job doing that!

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igloo myrtilles fourmis