Minimize this nasty expression!

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Anindya Biswas
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Minimize this nasty expression!

Unread post by Anindya Biswas » Tue Jan 12, 2021 12:46 pm

Let $a,b,c,d$ be positive real numbers such that $a+b+c+d=5$
Find the minimum value of $\sqrt{a^2+1}+\sqrt{b^2+4}+\sqrt{c^2+16}+\sqrt{d^2+25}$
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
John von Neumann

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Mehrab4226
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Re: Minimize this nasty expression!

Unread post by Mehrab4226 » Thu Jan 28, 2021 10:41 pm

Anindya Biswas wrote:
Tue Jan 12, 2021 12:46 pm
Let $a,b,c,d$ be positive real numbers such that $a+b+c+d=5$
Find the minimum value of $\sqrt{a^2+1}+\sqrt{b^2+4}+\sqrt{c^2+16}+\sqrt{d^2+25}$
Can we find the value or the minimum value of $(abcd)$ from $a+b+c+d=5$?? :?:
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré

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Mehrab4226
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Re: Minimize this nasty expression!

Unread post by Mehrab4226 » Thu Jan 28, 2021 11:44 pm

I couldn't get the solution yet. But I made a little progress, which may be no progress at all but still, I am presenting it.
By AM-GM we get,
$\frac{a^2+1}{2} \geq \sqrt{a^2 \times 1}$
$\therefore a^2+1 \geq 2a$
Similarly,
$b^2+4 \geq 4b$
$c^2+16 \geq 8c$
$d^2+25 \geq 10d$

Substituting this with the value we want to achieve and using AM-GM again we get,

$\frac{\sqrt{2a}+\sqrt{4b}+\sqrt{8c}+\sqrt{10d}}{4} \geq \sqrt[4]{\sqrt{640abcd}} $

$\therefore \sqrt{a^2+1}+\sqrt{b^2+4}+\sqrt{c^2+16}+\sqrt{d^2+25} \geq \sqrt{2a}+\sqrt{4b}+\sqrt{8c}+\sqrt{10d} \geq 4\sqrt[4]{\sqrt{640abcd}} $

Now if we can find the minimum of $(abcd)$ we are done.
NOTE: We haven't used the information $a+b+c+d=5$ yet. Probably to find the value of $(abcd)$ we need that information or this whole progress was just a wild goose chase
This may not be the right way to go the solution may be different!
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré

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Anindya Biswas
Posts:264
Joined:Fri Oct 02, 2020 8:51 pm
Location:Magura, Bangladesh
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Re: Minimize this nasty expression!

Unread post by Anindya Biswas » Fri Jan 29, 2021 6:07 am

Mehrab4226 wrote:
Thu Jan 28, 2021 11:44 pm
I couldn't get the solution yet. But I made a little progress, which may be no progress at all but still, I am presenting it.
By AM-GM we get,
$\frac{a^2+1}{2} \geq \sqrt{a^2 \times 1}$
$\therefore a^2+1 \geq 2a$
Similarly,
$b^2+4 \geq 4b$
$c^2+16 \geq 8c$
$d^2+25 \geq 10d$

Substituting this with the value we want to achieve and using AM-GM again we get,

$\frac{\sqrt{2a}+\sqrt{4b}+\sqrt{8c}+\sqrt{10d}}{4} \geq \sqrt[4]{\sqrt{640abcd}} $

$\therefore \sqrt{a^2+1}+\sqrt{b^2+4}+\sqrt{c^2+16}+\sqrt{d^2+25} \geq \sqrt{2a}+\sqrt{4b}+\sqrt{8c}+\sqrt{10d} \geq 4\sqrt[4]{\sqrt{640abcd}} $

Now if we can find the minimum of $(abcd)$ we are done.
NOTE: We haven't used the information $a+b+c+d=5$ yet. Probably to find the value of $(abcd)$ we need that information or this whole progress was just a wild goose chase
This may not be the right way to go the solution may be different!
There is a problem with the minimum value of $abcd$, notice that this value can be arbitrarily close to $0$.
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
John von Neumann

User avatar
Mehrab4226
Posts:230
Joined:Sat Jan 11, 2020 1:38 pm
Location:Dhaka, Bangladesh

Re: Minimize this nasty expression!

Unread post by Mehrab4226 » Fri Jan 29, 2021 2:51 pm

Anindya Biswas wrote:
Fri Jan 29, 2021 6:07 am
Mehrab4226 wrote:
Thu Jan 28, 2021 11:44 pm
I couldn't get the solution yet. But I made a little progress, which may be no progress at all but still, I am presenting it.
By AM-GM we get,
<span class="typeset"><nobr><span class="scale"><span style="display:inline-block; width:6.693em"><span style="position:relative;"><span style="position:absolute; top:0em; left:0em;"><span class="spacer" style="margin-left:0.12em"></span><span style="position: relative; margin-left:0.526em; top:0.344em;"><span class="size2"><span class="icmr10">2</span></span></span><span style="position: relative; margin-left:-0.902em; top:-0.393em;"><span class="size2"><span class="icmmi10">a</span></span><span style="position: relative; top:-0.254em;"><span class="size0"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.035em"></span></span><span class="size2"><span class="icmr10">+1</span></span></span><span style="position: relative; margin-left:-1.428em; top:-0.27em;"><span class="blank" style="border-left:1.428em solid;height:1.5px;"></span></span><span class="spacer" style="margin-left:0.12em"></span><span style="position: relative; margin-left:0.277em;"><img src="jsMath/fonts/cmsy10/alpha/100/char15.png" style=" width:10px; vertical-align:-2px; margin-right:0.058em;"></span><span style="position: relative; margin-left:0.277em;"><span style="position: relative; top:-0.984em;"><img src="jsMath/fonts/cmex10/alpha/100/char70.png" style=" width:15px; vertical-align:-17px; margin-right:-0.079em;"></span><span style="position: relative; top:-0.954em;"><span class="blank" style="border-left:2.632em solid;height:1.5px;"></span></span><span style="position: relative; margin-left:-2.632em;"><span class="icmmi10">a</span><span style="position: relative; top:-0.288em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.222em;"><img src="jsMath/fonts/cmsy10/alpha/100/char02.png" style=" width:9px; margin-right:0.130em;"></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">1</span></span></span></span> </span><span class="blank" style="height:0.827em;"></span></span></span><span class="blank" style="height:1.549em;vertical-align:-0.394em"></span></span></nobr></span>
<span class="typeset"><nobr><span class="scale"><img src="jsMath/fonts/msam10/alpha/100/char29.png" style=" width:9px; vertical-align:-2px; margin-right:0.019em;"><span style="position: relative; margin-left:0.277em;"><span class="icmmi10">a</span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">1</span></span><span style="position: relative; margin-left:0.277em;"><img src="jsMath/fonts/cmsy10/alpha/100/char15.png" style=" width:10px; vertical-align:-2px; margin-right:0.058em;"></span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">2</span></span><span class="icmmi10">a</span><span class="blank" style="height:1.126em;vertical-align:-0.186em"></span></span></nobr></span>
Similarly,
<span class="typeset"><nobr><span class="scale"><span class="icmmi10">b</span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">4</span></span><span style="position: relative; margin-left:0.277em;"><img src="jsMath/fonts/cmsy10/alpha/100/char15.png" style=" width:10px; vertical-align:-2px; margin-right:0.058em;"></span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">4</span></span><span class="icmmi10">b</span><span class="blank" style="height:1.126em;vertical-align:-0.186em"></span></span></nobr></span>
<span class="typeset"><nobr><span class="scale"><span class="icmmi10">c</span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">1</span></span><span class="icmr10">6</span><span style="position: relative; margin-left:0.277em;"><img src="jsMath/fonts/cmsy10/alpha/100/char15.png" style=" width:10px; vertical-align:-2px; margin-right:0.058em;"></span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">8</span></span><span class="icmmi10">c</span><span class="blank" style="height:1.126em;vertical-align:-0.186em"></span></span></nobr></span>
<span class="typeset"><nobr><span class="scale"><span class="icmmi10">d</span><span style="position: relative; top:-0.362em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">2</span></span><span class="icmr10">5</span><span style="position: relative; margin-left:0.277em;"><img src="jsMath/fonts/cmsy10/alpha/100/char15.png" style=" width:10px; vertical-align:-2px; margin-right:0.058em;"></span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">1</span></span><span class="icmr10">0</span><span class="icmmi10">d</span><span class="blank" style="height:1.126em;vertical-align:-0.186em"></span></span></nobr></span>

Substituting this with the value we want to achieve and using AM-GM again we get,

<span class="typeset"><nobr><span class="scale"><span style="display:inline-block; width:13.688em"><span style="position:relative;"><span style="position:absolute; top:0em; left:0em;"><span class="spacer" style="margin-left:0.12em"></span><span style="position: relative; margin-left:3.196em; top:0.344em;"><span class="size2"><span class="icmr10">4</span></span></span><span style="position: relative; margin-left:-3.572em; top:-0.462em;"><span style="position: relative; top:-0.599em;"><span class="size2"><img src="jsMath/fonts/cmsy10/alpha/70/char70.png" style=" width:9px; vertical-align:-10px; margin-right:-0.094em;"></span></span><span style="position: relative; top:-0.578em;"><span class="blank" style="border-left:0.676em solid;height:1.5px;"></span></span><span style="position: relative; margin-left:-0.676em;"><span class="size2"><span class="icmr10">2</span><span class="icmmi10">a</span></span></span><span class="size2"><span class="icmr10">+</span></span><span style="position: relative; top:-0.599em;"><span class="size2"><img src="jsMath/fonts/cmsy10/alpha/70/char70.png" style=" width:9px; vertical-align:-10px; margin-right:-0.094em;"></span></span><span style="position: relative; top:-0.578em;"><span class="blank" style="border-left:0.676em solid;height:1.5px;"></span></span><span style="position: relative; margin-left:-0.676em;"><span class="size2"><span class="icmr10">4</span><span class="icmmi10">b</span></span></span><span class="size2"><span class="icmr10">+</span></span><span style="position: relative; top:-0.626em;"><span class="size2"><img src="jsMath/fonts/cmsy10/alpha/70/char70.png" style=" width:9px; vertical-align:-10px; margin-right:-0.094em;"></span></span><span style="position: relative; top:-0.605em;"><span class="blank" style="border-left:0.676em solid;height:1.5px;"></span></span><span style="position: relative; margin-left:-0.676em;"><span class="size2"><span class="icmr10">8</span><span class="icmmi10">c</span></span></span><span class="size2"><span class="icmr10">+</span></span><span style="position: relative; top:-0.608em;"><span class="size2"><img src="jsMath/fonts/cmsy10/alpha/70/char70.png" style=" width:9px; vertical-align:-10px; margin-right:-0.094em;"></span></span><span style="position: relative; top:-0.587em;"><span class="blank" style="border-left:1.052em solid;height:1.5px;"></span></span><span style="position: relative; margin-left:-1.052em;"><span class="size2"><span class="icmr10">10</span><span class="icmmi10">d</span></span></span></span><span style="position: relative; margin-left:-6.768em; top:-0.27em;"><span class="blank" style="border-left:6.768em solid;height:1.5px;"></span></span><span class="spacer" style="margin-left:0.12em"></span><span style="position: relative; margin-left:0.277em;"><img src="jsMath/fonts/cmsy10/alpha/100/char15.png" style=" width:10px; vertical-align:-2px; margin-right:0.058em;"></span><span style="position: relative; margin-left:0.277em;"><span style="position: relative; top:-1.392em;"><img src="jsMath/fonts/cmex10/alpha/100/char71.png" style=" width:15px; vertical-align:-25px; margin-right:-0.079em;"></span><span style="position: relative; margin-left:-0.614em; top:-0.835em;"><span class="size0"><span class="icmr10">4</span></span></span><span style="position: relative; margin-left:0.388em; top:-1.362em;"><span class="blank" style="border-left:4.286em solid;height:1.5px;"></span></span><span style="position: relative; margin-left:-4.286em;"><span style="position: relative; top:-0.894em;"><img src="jsMath/fonts/cmsy10/alpha/100/char70.png" style=" width:12px; vertical-align:-14px; margin-right:-0.030em;"></span><span style="position: relative; top:-0.864em;"><span class="blank" style="border-left:3.459em solid;height:1.5px;"></span></span><span style="position: relative; margin-left:-3.459em;"><span class="icmr10">640</span><span class="icmmi10">abcd</span></span></span></span> </span><span class="blank" style="height:0.827em;"></span></span></span><span class="blank" style="height:1.98em;vertical-align:-0.417em"></span></span></nobr></span>

<span class="typeset"><nobr><span class="scale"><span style="display:inline-block; width:37.679em"><span style="position:relative;"><span style="position:absolute; top:0em; left:0em;"><img src="jsMath/fonts/msam10/alpha/100/char29.png" style=" width:9px; vertical-align:-2px; margin-right:0.019em;"><span style="position: relative; margin-left:0.277em;"><span style="position: relative; top:-0.925em;"><img src="jsMath/fonts/cmsy10/alpha/100/char70.png" style=" width:12px; vertical-align:-14px; margin-right:-0.030em;"></span><span style="position: relative; top:-0.895em;"><span class="blank" style="border-left:2.406em solid;height:1.5px;"></span></span><span style="position: relative; margin-left:-2.406em;"><span class="icmmi10">a</span><span style="position: relative; top:-0.288em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">1</span></span></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span style="position: relative; top:-0.925em;"><img src="jsMath/fonts/cmsy10/alpha/100/char70.png" style=" width:12px; vertical-align:-14px; margin-right:-0.030em;"></span><span style="position: relative; top:-0.895em;"><span class="blank" style="border-left:2.406em solid;height:1.5px;"></span></span><span style="position: relative; margin-left:-2.406em;"><span class="icmmi10">b</span><span style="position: relative; top:-0.288em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">4</span></span></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span style="position: relative; top:-0.925em;"><img src="jsMath/fonts/cmsy10/alpha/100/char70.png" style=" width:12px; vertical-align:-14px; margin-right:-0.030em;"></span><span style="position: relative; top:-0.895em;"><span class="blank" style="border-left:2.857em solid;height:1.5px;"></span></span><span style="position: relative; margin-left:-2.857em;"><span class="icmmi10">c</span><span style="position: relative; top:-0.288em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">1</span></span><span class="icmr10">6</span></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span style="position: relative; top:-0.925em;"><img src="jsMath/fonts/cmsy10/alpha/100/char70.png" style=" width:12px; vertical-align:-14px; margin-right:-0.030em;"></span><span style="position: relative; top:-0.895em;"><span class="blank" style="border-left:2.933em solid;height:1.5px;"></span></span><span style="position: relative; margin-left:-2.933em;"><span class="icmmi10">d</span><span style="position: relative; top:-0.288em;"><span class="size2"><span class="icmr10">2</span></span><span class="spacer" style="margin-left:0.05em"></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">2</span></span><span class="icmr10">5</span></span></span><span style="position: relative; margin-left:0.277em;"><img src="jsMath/fonts/cmsy10/alpha/100/char15.png" style=" width:10px; vertical-align:-2px; margin-right:0.058em;"></span><span style="position: relative; margin-left:0.277em;"><span style="position: relative; top:-0.867em;"><img src="jsMath/fonts/cmsy10/alpha/100/char70.png" style=" width:12px; vertical-align:-14px; margin-right:-0.030em;"></span><span style="position: relative; top:-0.837em;"><span class="blank" style="border-left:0.977em solid;height:1.5px;"></span></span><span style="position: relative; margin-left:-0.977em;"><span class="icmr10">2</span><span class="icmmi10">a</span></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span style="position: relative; top:-0.867em;"><img src="jsMath/fonts/cmsy10/alpha/100/char70.png" style=" width:12px; vertical-align:-14px; margin-right:-0.030em;"></span><span style="position: relative; top:-0.837em;"><span class="blank" style="border-left:0.977em solid;height:1.5px;"></span></span><span style="position: relative; margin-left:-0.977em;"><span class="icmr10">4</span><span class="icmmi10">b</span></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span style="position: relative; top:-0.894em;"><img src="jsMath/fonts/cmsy10/alpha/100/char70.png" style=" width:12px; vertical-align:-14px; margin-right:-0.030em;"></span><span style="position: relative; top:-0.864em;"><span class="blank" style="border-left:0.977em solid;height:1.5px;"></span></span><span style="position: relative; margin-left:-0.977em;"><span class="icmr10">8</span><span class="icmmi10">c</span></span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span style="position: relative; top:-0.894em;"><img src="jsMath/fonts/cmsy10/alpha/100/char70.png" style=" width:12px; vertical-align:-14px; margin-right:-0.030em;"></span><span style="position: relative; top:-0.864em;"><span class="blank" style="border-left:1.504em solid;height:1.5px;"></span></span><span style="position: relative; margin-left:-1.504em;"><span class="icmr10">10</span><span class="icmmi10">d</span></span></span><span style="position: relative; margin-left:0.277em;"><img src="jsMath/fonts/cmsy10/alpha/100/char15.png" style=" width:10px; vertical-align:-2px; margin-right:0.058em;"></span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">4</span></span><span style="position: relative; top:-1.392em;"><img src="jsMath/fonts/cmex10/alpha/100/char71.png" style=" width:15px; vertical-align:-25px; margin-right:-0.079em;"></span><span style="position: relative; margin-left:-0.614em; top:-0.835em;"><span class="size0"><span class="icmr10">4</span></span></span><span style="position: relative; margin-left:0.388em; top:-1.362em;"><span class="blank" style="border-left:4.286em solid;height:1.5px;"></span></span><span style="position: relative; margin-left:-4.286em;"><span style="position: relative; top:-0.894em;"><img src="jsMath/fonts/cmsy10/alpha/100/char70.png" style=" width:12px; vertical-align:-14px; margin-right:-0.030em;"></span><span style="position: relative; top:-0.864em;"><span class="blank" style="border-left:3.459em solid;height:1.5px;"></span></span><span style="position: relative; margin-left:-3.459em;"><span class="icmr10">640</span><span class="icmmi10">abcd</span></span></span> </span><span class="blank" style="height:0.827em;"></span></span></span><span class="blank" style="height:1.98em;vertical-align:-0.417em"></span></span></nobr></span>

Now if we can find the minimum of <span class="typeset"><nobr><span class="scale"><span class="icmr10">(</span><span class="icmmi10">abcd</span><span class="icmr10">)</span></span></nobr></span> we are done.
NOTE: We haven't used the information <span class="typeset"><nobr><span class="scale"><span class="icmmi10">a</span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">b</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">c</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmr10">+</span></span><span style="position: relative; margin-left:0.222em;"><span class="icmmi10">d</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">=</span></span><span style="position: relative; margin-left:0.277em;"><span class="icmr10">5</span></span></span></nobr></span> yet. Probably to find the value of <span class="typeset"><nobr><span class="scale"><span class="icmr10">(</span><span class="icmmi10">abcd</span><span class="icmr10">)</span></span></nobr></span> we need that information or this whole progress was just a wild goose chase
This may not be the right way to go the solution may be different!
There is a problem with the minimum value of <span class="typeset"><nobr><span class="scale"><span class="icmmi10">abcd</span></span></nobr></span>, notice that this value can be arbitrarily close to <span class="typeset"><nobr><span class="scale"><span class="icmr10">0</span></span></nobr></span>.
That is why I said this process can be wrong.
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré

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