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

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

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

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:

$$f_{\min}=\frac{13}{3}$$
 
MarkFL said:
My solution:

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!
 
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