SOLUTION: What is the minimum product of two numbers whose difference is 5? What are the numbers?

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Question 1193027: What is the minimum product of two numbers whose difference is 5? What are the numbers?
Answer by ikleyn(52777)   (Show Source): You can put this solution on YOUR website!
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What is the minimum product of two numbers whose difference is 5? What are the numbers?
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Let x and y be the numbers; let x be the lesser number and y be the greater number, so y-x = 5.


Consider the mean value a =   and half the distance between x and y,  b =  =  = 2.5


Then, obviously,  x = a - 2.5  and  y = a + 2.5, so we can write


    xy = (a-2.5)*(a+2.5) = .


From this formula, it is clear that xy is minimum when a= 0, i.e. when x+y = 0;  in other words, the numbers x and y 
shoud be opposite: y = -x.


Thus we get that the minimum of the product xy  is achieved when x = -y.


Together with y-x = 5 it implies that y - (-y) = 5,  or  2y = 5, y = 5/2 = 2.5.


Thus the minimum value of xy is  at x= -2.5,  y= 2.5,  and the value of xy is   = -6.25 then.   


ANSWER.  The minimum is achieved when x= -2.5, y= 2.5.

         The minimum value of the product is  then -6.25.

Solved.

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There are many different ways and methods of solving such problems.

They differ in the level of necessary prerequisites, which the reader should know to understand the solution.

What I exposed here, is traditionally considered in Math and in Math popular literature as
the SIMPLEST method of solving, accessible for students with minimal level of knowledge (for the most young readers).



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