Let x, y, and z be positive real numbers. Find the minimum value of![](/api/ssl-img-proxy?src=https%3A%2F%2Flatex.artofproblemsolving.com%2Ff%2Ff%2F3%2Fff35e4015812d97206d31d9bfc733db67587e13a.png)


[Help Please!!!](/questions/help-please_60819)
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By Cauchy-Schwarz, the minimum value is (1 + 2 + 4)^2 = 49.


a=1; b=1;c=1;p=39;d=(a +2*b+ 4*c) *(4/a + 2/b + 1/c);if(d==36, goto6, goto7);printd, a, b, c;p=p-1;if(p>=1, goto4,0); a++;if(a<50, goto4, 0);a=1;b++;if(b<50, goto4, discard=0; a=1;b=1;c++;if(c<50, goto4, 0)

  

 The minimum value =36 and x, y, z take on an infinite number of values. Here are just a few:

 mv   x y z

 36    4 2 1  
 36    8 4 2  
 36    12 6 3  
 36    16 8 4  
 36    20 10 5  
 36    24 12 6  
 36    28 14 7  
 36    32 16 8  
 36    36 18 9  
 36    40 20 10  
 36    44 22 11  
 36    48 24 12  
  

