Find Min Value: $a,b,c>0$ with $a+b+c=k$

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The discussion focuses on minimizing the expression $\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{c^2+a^2}$ under the constraints $a, b, c > 0$ and $a + b + c = k$. Participants confirm that the problem is well-defined and agree on the necessity of applying optimization techniques to find the minimum value. The solution involves leveraging inequalities and possibly calculus to derive the optimal values of $a$, $b$, and $c$ that satisfy the given conditions.

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$a,b,c>0$

$a+b+c=k$

find:$min(\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{c^2+a^2})$
 
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My Solution:

Given $$a+b+c = k$$ and $$a,b,c>0$$

Now we can write $$\sqrt{a^2+b^2} = \left|a+ib\right|$$ and $$\sqrt{b^2+c^2} = \left|b+ic\right|$$ and $$\sqrt{c^2+a^2} = \left|c+ia\right|$$

Where $$i=\sqrt{-1}$$ So Using Triangle Inequality of Complex number

$$\left|a+ib\right|+\left|b+ic\right|+\left|c+ia\right|\geq \left|\left(a+b+c\right)+i\left(b+c+a\right)\right| = \left|k+ik\right|=\sqrt{2}k$$

and equality hold when $$\displaystyle \frac{a}{b} = \frac{b}{c} = \frac{c}{a}$$
 
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jacks said:
My Solution:

Given $$a+b+c = k$$ and $$a,b,c>0$$

Now we can write $$\sqrt{a^2+b^2} = \left|a+ib\right|$$ and $$\sqrt{b^2+c^2} = \left|b+ic\right|$$ and $$\sqrt{c^2+a^2} = \left|c+ia\right|$$

Where $$i=\sqrt{-1}$$ So Using Triangle Inequality of Complex number

$$\left|a+ib\right|+\left|b+ic\right|+\left|c+ia\right|\geq \left|\left(a+b+c\right)+i\left(b+c+a\right)\right| = \left|k+ik\right|=\sqrt{2}k$$

and equality hold when $$\displaystyle \frac{a}{b} = \frac{b}{c} = \frac{c}{a}$$
nice solution !
 
Albert said:
$a,b,c>0$

$a+b+c=k$

find:$min(\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{c^2+a^2})$
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