2

A few weeks ago I took part in a math competition and had this question through some trial and error I got the correct answer 80 as I didn't have time to use maximization using calculus, is there a way to do this question quickly in a rigorous way:

Find the maximum possible value of $$ 9\sqrt{x}+8\sqrt{y}+5\sqrt{z} $$

where x, y, and z are positive real numbers satisfying $$ 9x +4y+z=128 $$

Any help would be greatly appreciated, thanks.

RobPratt
  • 45,619
Il Leone
  • 41
  • 2
    Do you know about Lagrange multipliers? – saulspatz May 07 '20 at 15:04
  • You mean setting partial derivatives equal to each other different by a scale factor of lambda right. I have done such questions with two variables before. I am only 15 so I have not been able to pursue such subjects in calculus 2 class. – Il Leone May 07 '20 at 15:09
  • Yes, that's what I mean, though the description isn't quite right. That should work quickly in this case. – saulspatz May 07 '20 at 15:11
  • I worked it out using Lagrange multipliers. The solution is <64/25, 256/25, 64> which ends up with the correct answer of 80. Thank you so much for your help! – Il Leone May 07 '20 at 15:20
  • It was my pleasure. – saulspatz May 07 '20 at 15:24

1 Answers1

2

Rename $(x,y,z)$ to $(x_1,x_2,x_3)$, and let $(c_1,c_2,c_3)=(9,8,5)$ and $(a_1,a_2,a_3)=(9,4,1)$. By Cauchy-Schwarz: \begin{align} \left(\sum_j c_j \sqrt{x_j}\right)^2 &= \left(\sum_j \frac{c_j}{\sqrt{a_j}} \sqrt{a_j x_j}\right)^2 \\ &\le \left(\sum_j \left(\frac{c_j}{\sqrt{a_j}}\right)^2\right) \left(\sum_j \sqrt{a_j x_j}^2\right)\\ &= \left(\sum_j \frac{c_j^2}{a_j}\right) \left(\sum_j a_j x_j\right)\\ &= \left(\frac{9^2}{9} + \frac{8^2}{4} + \frac{5^2}{1}\right) 128\\ &= 6400, \end{align} so $$\sum_j c_j \sqrt{x_j} \le 80$$

RobPratt
  • 45,619