# algebra problem

• Jan 28th 2013, 11:57 PM
Swarnav
algebra problem
a,b>0, a+b=1, then prove that (a^2b)+(b^2a)<=1
• Jan 29th 2013, 12:23 AM
topsquark
Re: algebra problem
Quote:

Originally Posted by Swarnav
a,b>0, a+b=1, then prove that (a^2b)+(b^2a)<=1

If a + b = 1 then we know that b = 1 - a. So we have
$\displaystyle a^2b + b^2 a = a^2 (1 - a) - (1 - a)^2$

Simplifying we eventually get
$\displaystyle a^2b + b^2 a = a^2 - a$

Now, you can probably find a more elegant way to prove that $\displaystyle -a^2 + a \leq 1$, but the simplest way to approach this is to simply graph it.

-Dan
• Jan 29th 2013, 12:47 AM
Swarnav
Re: algebra problem
it is not (a^2)*b or(b^2)a,it is a^(2b)+b^(2a)
• Feb 7th 2013, 01:35 AM
Swarnav
Re: algebra problem
please someone help me to solve it.
• Feb 7th 2013, 03:53 AM
pssingh1001
Re: algebra problem
a,b>0, a+b=1, then prove that (a^2b)+(b^2a)<=1

a+ b =1 are given
then
a= 1-b
multiply a both side

a^2 = a -a*b
multiply b both side
a^2 *b = ab -a* b^2
then
a^2 *b + b^2 *a = ab
a,b>0 & a+b =1 are given
then ab>=1

then we can say that
(a^2b)+(b^2a)<=1

hence proved
• Feb 7th 2013, 03:56 AM
pssingh1001
Re: algebra problem
I think this is the right ans...............
if u know something diff..............
plz share with whole team.........
Regards
prem
• Feb 8th 2013, 03:04 AM
CosmoBlazer
Re: algebra problem
a^2b+b^2a=ab(a+b)=ab
according to the basic inequelity:a>0,b>0,so√ab≤（a+b）/2=1/2
we see the original polynomial=ab=(√ab)^2≤1/4
the restriction of problem is so loose that we can suspect there are some other ways to solve it
• Feb 15th 2013, 09:33 PM
pssingh1001
Re: algebra problem
Hi CosmoBlazer !
This solution is too good.I did not think such type of logic .....
this is the more easy and time less solution...............

thanks
• Feb 15th 2013, 09:35 PM
pssingh1001
Re: algebra problem
if u have something other easiest evaluation ..

thanks
• Feb 16th 2013, 03:29 AM
ibdutt
Re: algebra problem
Quote:

Originally Posted by pssingh1001
a,b>0, a+b=1, then prove that (a^2b)+(b^2a)<=1

a+ b =1 are given
then
a= 1-b
multiply a both side

a^2 = a -a*b
multiply b both side
a^2 *b = ab -a* b^2
then
a^2 *b + b^2 *a = ab
a,b>0 & a+b =1 are given
then ab>=1

then we can say that
(a^2b)+(b^2a)<=1

hence proved

When a + b = 1 and a,b >0 then ab <=0
Then we can conclude that (a^2b)+(b^2a)<=1