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- Feb 14, 2012

- 3,802

- Thread starter anemone
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- #1

- Feb 14, 2012

- 3,802

- Nov 26, 2013

- 732

Given the conditions:

$a \leq 1 \;\;\wedge \;\;a+b \leq 5\;\; \wedge \;\;a+b+c\leq 14\;\; \wedge \;\; a+b+c+d \leq 30$

which by successive subtractions implies:

$a \leq 1 \;\;\wedge \;\;b \leq 4\;\; \wedge \;\;c\leq 9\;\; \wedge \;\; d \leq 16$

or:

$\sqrt{a} \leq 1 \;\;\wedge \;\;\sqrt{b} \leq 2\;\; \wedge \;\;\sqrt{c}\leq 3\;\; \wedge \;\; \sqrt{d} \leq 4$

Adding the four inequalities yields:

$\sqrt{a} +\sqrt{b} +\sqrt{c}+\sqrt{d} \leq 1+2+3+4 = 10.$

- Mar 31, 2013

- 1,331

Given the conditions:

$a \leq 1 \;\;\wedge \;\;a+b \leq 5\;\; \wedge \;\;a+b+c\leq 14\;\; \wedge \;\; a+b+c+d \leq 30$

which by successive subtractions implies:

$a \leq 1 \;\;\wedge \;\;b \leq 4\;\; \wedge \;\;c\leq 9\;\; \wedge \;\; d \leq 16$

or:

$\sqrt{a} \leq 1 \;\;\wedge \;\;\sqrt{b} \leq 2\;\; \wedge \;\;\sqrt{c}\leq 3\;\; \wedge \;\; \sqrt{d} \leq 4$

Adding the four inequalities yields:

$\sqrt{a} +\sqrt{b} +\sqrt{c}+\sqrt{d} \leq 1+2+3+4 = 10.$

a = .2 b = 4.6 does not satisfy your consideration.

- Nov 26, 2013

- 732

Thankyou for your comment. You´re right of course. I did miss something ...

- Apr 4, 2014

- 66

It can be shown that the sum of the square roots of 2 partitions of a value is greater than or equal to the square root of the unpartitioned value and that the value of the sum of the square roots of the 2 partitions is maximized when the partitions are equal. The equation is:

$\sqrt{a}+\sqrt{b} = \sqrt{a+b+2\sqrt{ab}}$

let the value being partitioned = 1 then a+b=1 and b=a-1 and the equation becomes:

$\sqrt{a}+\sqrt{b} = \sqrt{1 + 2\sqrt{a(1-a)}}$

and it can be seen that as $a$ ranges from 0 to 1, the maximum value of the sum of the square roots of the two partitions is at $a$ = 1/2. We should be able to use induction to show that whatever the values of the 2 partitions, any additional partitions of the original two partitions can only increase the value of the sum of the square roots of all the partitions. Applying this to the constraining inequalities given for the four partitions we find that

a=1, b=4, c=9 and d=16 give the maximal value for the sum of the square roots of these four partitions within the given constraints and that this sum is 10 which demonstrates the assertion.

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- Feb 14, 2012

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$k_1f(x_1)+k_2f(x_2)+\cdots+k_nf(x_n)\le f(k_1x_1+k_2x_2+\cdots+k_nx_n)$

Now, take $n=4$ and $k_1=\dfrac{1}{10},\,k_2=\dfrac{2}{10},\,k_3=\dfrac{3}{10},\,k_4=\dfrac{4}{10}$. It follows that

$\dfrac{1}{10}\sqrt{a}+\dfrac{2}{10}\sqrt{\dfrac{b}{4}}+\dfrac{3}{10}\sqrt{\dfrac{c}{9}}+\dfrac{4}{10}\sqrt{\dfrac{d}{16}}\le \sqrt{\dfrac{a}{10}+\dfrac{b}{20}+\dfrac{c}{30}+\dfrac{d}{40}}$

or

$\sqrt{a}+\sqrt{b}+\sqrt{c}+\sqrt{d}\le 10\sqrt{\dfrac{12a+6b+4c+3d}{120}}$

But

$12a+6b+4c+3d=3(a+b+c+d)+(a+b+c)+2(a+b)+6a\le 3(30)+14+2(5)+6(1)=120$

and the claim is then proved.