Find the minimum of (4xyz)/(3)+x²+y²+z²

[anemone]

Let $x,\,y,\,z$ be the lengths of the sides of a triangle such that $x+y+z=3$.

Find the minimum of $\dfrac{4xyz}{3}+x^2+y^2+z^2$.

 

[MarkFL]

My solution:

Spoiler

We see that there is cyclic symmetry in the objective function $f$ and constraint $g$, thus the critical point is:

\(\displaystyle (x,y,z)=(1,1,1)\)

We then find:

\(\displaystyle f(1,1,1)=\frac{13}{3}\)

Testing another point:

\(\displaystyle f\left(\frac{1}{2},\frac{1}{2},2\right)>\frac{13}{3}\)

Hence:

\(\displaystyle f_{\min}=\frac{13}{3}\)

 

[anemone]

My solution:

Spoiler

We see that there is cyclic symmetry in the objective function $f$ and constraint $g$, thus the critical point is:

\(\displaystyle (x,y,z)=(1,1,1)\)

We then find:

\(\displaystyle f(1,1,1)=\frac{13}{3}\)

Testing another point:

\(\displaystyle f\left(\frac{1}{2},\frac{1}{2},2\right)>\frac{13}{3}\)

Hence:

\(\displaystyle f_{\min}=\frac{13}{3}\)

Very well done, MarkFL! (Sun) And thanks for participating!