Fascinating Maxima

Algebra Level 4

2 x + 3 y + 4 z + 5 a \large 2\sqrt x + 3\sqrt y + 4\sqrt z + 5\sqrt a

Let x , y , z x,y,z and a a be real numbers satisfying 4 x + 9 y + 16 z + 25 a = 720 4x+9y+16z+25a=720 , find the maximum value of the expression above.

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The answer is 53.66.

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Akshay Sharma by Akshay Sharma , 21, India

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1 solution

We know that x , y , z , a ≥ 0 x,y,z,a \ge 0
Applyning Cauchy-Schwarz Inequality,
( 4 x + 9 y + 16 z + 25 a ) ( 1 2 + 1 2 + 1 2 + 1 2 ) ≥ 2 x + 3 y + 4 z + 5 a \sqrt{(4x+9y+16z + 25a)(1^{2}+1^{2}+1^{2}+1^{2})} \ge 2 \sqrt x+3\sqrt y+4\sqrt z+5\sqrt a
2 x + 3 y + 4 z + 5 a ≤ 2880 ≈ 53.66 2 \sqrt x+3\sqrt y+4\sqrt z+5\sqrt a \le \sqrt{2880} \approx 53.66


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Moderator note:

To prove that it is indeed the maximum, you have to show that it can be achieved. Otherwise, we only have an upper bound.

Equality occurs when,
4 x = 9 y = 16 z = 25 a = 180 4x = 9y = 16z = 25a = 180
x = 45 , y = 20 , z = 11.25 , a = 7.2 x = 45, y = 20, z = 11.25, a = 7.2 which indeed satisfy both conditions.

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