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Of all numbers x and y whose sum is​ 50, the two that have the maximum product are xequals=25 and yequals=25. that​ is, if xplus+yequals=​50, then xequals=25 and yequals=25 maximize qequals=xy. can there be a minimum​ product? why or why​ not?

Respuesta :

barnuts

x + y = 50 

Q = x * y 

 

Combining the two:
Q = x * (50 - x) 
Q = 50x - x^2 
Q = -x^2 + 50x 
Q = -(x^2 - 50x) 
Q = -(x^2 - 2 * 25x + 25^2 - 25^2) 
Q = 25^2 - (x - 25)^2 
Q = 625 - (x - 25)^2 

So what we got is an equation for parabola with a vertex at (25 , 625) and it opens downward. We know that parabolas only have one critical value, so if x and y are unrestricted, then there's no minimum product.