<s>[INST] maximum value of $xy+yz+zx.$ given $x+2y+z=4$ if $x+2y+z=4$ and $x,y,z$ are real number. then find maximum value of $xy+yz+zx$
putting $x+z=4-2z$ in $y(x+z)+zx = y(4-2z)+zx = 4y-2yz+zx$
i wan,t be able to  go further,could some help me with this 
 [/INST] $$xy+yz+zx+y^2 = (x+y)(y+z) \le \frac{(x+2y+z)^2}{4} = 4$$
$$\Rightarrow xy+yz+zx \le 4-y^2 \le 4$$
Equality occurs when $y=0$, $x=z=2$
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