Geometry or Algebra or Calculus?

Geometry Level 3

Given x , y , z x,y,z are nonnegative real values, such that x + y + z = 3 x + y + z = 3 , find the maximum value of 4 x y + 8 y z + 6 x z 4xy + 8yz + 6xz .

Challenge: \text{Challenge:} Solve this problem without any calculus.


The answer is 18.7826.

Michael Huang by Michael Huang , 29, USA

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1 solution

Pi Han Goh
Mar 28, 2021

Substitute z = 3 − x − y z = 3-x-y into the expression in question yields F : = − 8 y 2 + 24 y − 10 x y − 6 x 2 + 18 x . F := -8y^2 + 24y - 10xy - 6x^2 + 18x .

By completing the square , F = − 8 y 2 + 24 y − 10 x y − 6 x 2 + 18 x 0 = − 2 ( 4 y 2 − 12 y + 5 x y ) − 6 x 2 + 18 x 0 = − 2 [ ( 2 y − 3 + 5 4 x ) 2 − 9 + 15 2 x − 25 16 x 2 ] − 6 x 2 + 18 x 0 = − 2 ( 2 y + 5 4 x − 3 ) 2 − 23 8 ( x 2 − 24 23 x − 144 23 ) 0 = − 2 ( 2 y + 5 4 x − 3 ) 2 − 23 8 [ ( x − 12 23 ) 2 − 1 2 2 2 3 2 − 144 23 ] 0 = − 2 ( 2 y + 5 4 x − 3 ) 2 ⏟ ⩾ 0 − 23 8 ( x − 12 23 ) 2 ⏟ ⩾ 0 + 432 23 ⩽ 432 23 ≈ 18.783 \begin{array} { r c l} F &=& -8y^2 + 24y - 10xy - 6x^2 + 18x \\ \phantom 0 \\ &=& -2(4y^2 - 12y + 5xy) - 6x^2 + 18x \\ \phantom 0 \\ &=& -2 \bigg [ \left(2y - 3 + \dfrac54 x \right)^2 - 9 + \dfrac{15}2 x - \dfrac{25}{16}x^2 \bigg ] - 6x^2 + 18x \\ \phantom 0 \\ &=& -2\left(2y + \dfrac54 x - 3\right)^2 - \dfrac{23}8 \left(x^2 - \dfrac{24}{23}x - \dfrac{144}{23} \right) \\ \phantom 0 \\ &=& -2\left(2y + \dfrac54 x - 3\right)^2 - \dfrac{23}8 \bigg [ \left( x - \dfrac{12}{23}\right)^2 - \dfrac{12^2}{23^2} - \dfrac{144}{23} \bigg ] \\ \phantom 0 \\ &=& -2\underbrace{\left(2y + \dfrac54 x - 3\right)^2}_{\geqslant \, 0} - \dfrac{23}{8} \underbrace{\left( x - \dfrac{12}{23}\right)^2}_{\geqslant \, 0 } + \dfrac{432}{23} \leqslant \dfrac{432}{23} \approx \boxed{18.783} \end{array}

The maximum occurs when 2 y − 5 4 x − 3 = x − 12 23 = 0 ⇔ ( x , y , z ) = ( 12 23 , 27 23 , 30 23 ) . 2y - \frac54x -3 = x - \frac{12}{23} = 0 \hspace{10pt} \Leftrightarrow \hspace{10pt} (x,y,z) = \left( \dfrac{12}{23}, \dfrac{27}{23}, \dfrac{30}{23} \right) .

2 Helpful 2 Interesting 3 Brilliant 0 Confused

You're Da Man when it comes to Completing-the -Square, Pi Han!

tom engelsman - 2 months, 2 weeks ago

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The true algebra master is here!

Michael Huang - 2 months, 2 weeks ago

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THANKS FOR THE COMPLIMENTS! THANKS FOR THE COMPLIMENTS!

Pi Han Goh - 2 months, 2 weeks ago

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