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Algebra Level 5

x y + y z + x z − 2 x y z \large xy+yz+xz - 2xyz

If x , y x,y and z z are non-negative real numbers satisfying x + y + z = 1 x+y+z=1 , find the maximum value of the expression above.

Give your answer to 3 decimal places.

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The answer is 0.259.

Akshay Sharma by Akshay Sharma , 21, India

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1 solution

Denote the expression by P = x y + y z + z x − 2 x y z \text{Denote the expression by P = }xy+yz+zx-2xyz

P + 2 = 1 + x + y + z + x y + y z + z x + x y z − 3 x y z = ( 1 + x ) ( 1 + y ) ( 1 + z ) − 3 x y z \begin{aligned}P+2=1+x+y+z+xy+yz+zx+xyz-3xyz = (1+x)(1+y)(1+z)-3xyz\end{aligned}

( 1 + x ) + ( 1 + y ) + ( 1 + z ) 3 ≥ ( ( 1 + x ) ( 1 + y ) ( 1 + z ) ) 1 3 \frac{(1+x)+(1+y)+(1+z)}{3}\ge ((1+x)(1+y)(1+z))^{\frac{1}{3}}

( 1 + x ) ( 1 + y ) ( 1 + z ) ≤ 64 27 (1+x)(1+y)(1+z)\le \frac{64}{27} , P ≤ 64 27 − 3 x y z − 2 P\le \frac{64}{27}-3xyz-2 & we have x y z ≤ 1 27 xyz\le \frac{1}{27} , so P ≤ 64 27 − 3 27 − 2 = 7 27 = 0.259 (Approx.) P\le \frac{64}{27}-\frac{3}{27}-2=\frac{7}{27}=0.259\text{(Approx.)}

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