Mersmerizing Minimizing Mania

Algebra Level 5

x + y + z + 20 x + z + 20 y + 2 x+y+z+\frac{20}{\sqrt{x+z}}+\frac{20}{\sqrt{y+2}}

Given that x , y x,y and z z are positive reals, find the minimum of the expression above.

Inspiration .


The answer is 25.85.

Harsh Shrivastava by Harsh Shrivastava , 20, India

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1 solution

Rishabh Jain
Feb 14, 2016

Just a simple application of AM-GM..write it as B = ( x + z ) + ( y + 2 ) + 10 x + z + 10 x + z + 10 y + 2 + 10 y + 2 − 2 \mathfrak{B}=\color{forestgreen}{(x+z)+(y+2)+\frac{10}{\sqrt{x+z}}+\frac{10}{\sqrt{x+z}}+\frac{10}{\sqrt{y+2}}+\frac{10}{\sqrt{y+2}}}-2 Apply A M ≥ G M \large\color{#D61F06}{AM\geq GM} B ≥ 6 ( 1 0 4 ) 1 6 − 2 ≈ 25.85 \large\mathfrak{B}\geq 6(10^4)^{\small{\frac{1}{6}}}-2\approx \boxed{25.85} For equality y = ( 10 ) 2 3 − 2 , x + z = ( 10 ) 2 3 \color{#3D99F6}{y=(10)^{\small{\frac{2}{3}}}-2, ~x+z=(10)^{\small{\frac{2}{3}}}} .

5 Helpful 0 Interesting 0 Brilliant 0 Confused

Nice solution, I was about to post a solution but you posted it before :P

Harsh Shrivastava - 5 years, 4 months ago

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