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

  • Context: MHB 
  • Thread starter Thread starter Albert1
  • Start date Start date
  • Tags Tags
    Minimum Value
Click For Summary

Discussion Overview

The discussion centers around finding the minimum value of the expression $\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{c^2+a^2}$ under the constraints that $a, b, c > 0$ and $a + b + c = k$. The scope includes mathematical reasoning and optimization techniques.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • Some participants present solutions to the problem, although the details of these solutions are not specified.
  • There is an expression of appreciation for one of the solutions, indicating that at least one proposed approach is well-received.
  • The problem is reiterated by another participant, suggesting a focus on the same mathematical challenge.

Areas of Agreement / Disagreement

The discussion does not show clear consensus, as multiple participants have proposed solutions without resolving which, if any, is definitive.

Contextual Notes

Details of the proposed solutions are not provided, leaving the discussion open to interpretation and further exploration of methods.

Albert1
Messages
1,221
Reaction score
0
$a,b,c>0$

$a+b+c=k$

find:$min(\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{c^2+a^2})$
 
Mathematics news on Phys.org
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 $$\displaystyle \frac{a}{b} = \frac{b}{c} = \frac{c}{a}$$
 
Last edited by a moderator:
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:$min(\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{c^2+a^2})$
View attachment 3308
 

Attachments

  • minimum (AB+BC+EF).jpg
    minimum (AB+BC+EF).jpg
    18.2 KB · Views: 100

Similar threads

MHB Find Min Value of a+b+c=1: $\sqrt {a^2+b^2}+\sqrt {b^2+c^2}+\sqrt {c^2+a^2}$
  • · Replies 2 ·
Replies
2
Views
1K
MHB Finding Min Value of $\dfrac{|b|+|c|}{a}$ from Roots of Cubic Equations
  • · Replies 4 ·
Replies
4
Views
2K
MHB Min Value of $\dfrac{a+3c}{a+2b+c}$+$\dfrac{4b}{a+b+2c}$+$\dfrac{8c}{a+b+3c}$
  • · Replies 1 ·
Replies
1
Views
2K
MHB Prove $\sqrt{a}+\sqrt{b}+\sqrt{c}+\sqrt{d}\le 10$ w/ $a,b,c,d>0$
  • · Replies 5 ·
Replies
5
Views
2K
MHB Finding $b(a+c)$ for Real Roots of $\sqrt{2014}x^3-4029x^2+2=0$
  • · Replies 1 ·
Replies
1
Views
1K
MHB Prove Triangle Inequality: $\sqrt{2}\sin A-2\sin B+\sin C=0$
  • · Replies 2 ·
Replies
2
Views
2K
MHB What is the value of $(a+c)(b+c)(a-d)(b-d)$ given specific roots?
  • · Replies 6 ·
Replies
6
Views
1K
MHB What Is the Minimum Value of $f$ with Given Conditions?
  • · Replies 2 ·
Replies
2
Views
1K
MHB Find Product: $(a^4+b^4+c^4)$ and $(a^6+b^6+c^6)$
  • · Replies 3 ·
Replies
3
Views
1K
MHB Min Value of $a^2+b^2$ in Quadratic Equation
  • · Replies 1 ·
Replies
1
Views
1K