Junior 2006/3

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tanmoy
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Junior 2006/3

Unread post by tanmoy » Thu Jan 30, 2014 10:49 pm

Prove that the product of two real numbers is maximum when the numbers are equal to each other while $a+b=$constant.
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asif e elahi
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Re: Junior 2006/3

Unread post by asif e elahi » Thu Jan 30, 2014 10:52 pm

Use AM-GM inequality.

tanmoy
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Re: Junior 2006/3

Unread post by tanmoy » Thu Jan 30, 2014 11:25 pm

sorry,I have got a mistake.There is (a+b) after while.This is: while a+b=constant
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tanmoy
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Re: Junior 2006/3

Unread post by tanmoy » Thu Jan 30, 2014 11:32 pm

Please give the solution
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asif e elahi
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Re: Junior 2006/3

Unread post by asif e elahi » Fri Jan 31, 2014 10:52 am

Let $a+b=m$.Then $ab\leq (\frac{a+b}{2})^{2}=\frac{m^{2}}{4}$. Equality holds when $a=b$.

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Fatin Farhan
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Re: Junior 2006/3

Unread post by Fatin Farhan » Fri Jan 31, 2014 1:53 pm

Can also be done without using anything.
$$b=m-a$$
$$ab=a(m-a)=am-a^2$$
$$=\frac{m^2}{4} -a^2+ 2\frac{m}{2}a-\frac{m^2}{4}$$
$$=\frac{m^2}{4}- (a-\frac{m}{2})^2$$.
So, ab will be maximum if
$$a-\frac{m}{2}=0$$
$$a=\frac{m}{2}$$.
$$b=m-a=\frac{m}{2}$$
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