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:

$$(x,y,z)=(1,1,1)$$

We then find:

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

Testing another point:

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

Hence:

$$f_{\min}=\frac{13}{3}$$

 

[anemone]

My solution:

Spoiler

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

$$(x,y,z)=(1,1,1)$$

We then find:

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

Testing another point:

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

Hence:

$$f_{\min}=\frac{13}{3}$$

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