SOLUTION: Find two numbers differing by 48 whose product is as small as possible.

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Question 1180887: Find two numbers differing by 48 whose product is as small as possible.
Found 2 solutions by ikleyn, MathLover1:
Answer by ikleyn(52905)   (Show Source): You can put this solution on YOUR website!
.

If x is the smaller number, the the greater number is (x+48), according to the condition,

and the product of the two numbers is  x*(x+48) = x^2 + 48.


They want you find the MINIMUM of this quadratic function.


The minimum is the vertex of the parabola, and on x-axis, this minimum is located
exactly half way between the zeroes of this quadratic function.


The zeroes are -48 and 0, so   =  = -24.


The minimal value of the function is  y = (-24)*(-24+48) = (-24)*24 = -576.


ANSWER.  The minimum value of the producr of such two numbers is -576.

         It is achieved at the pair of the numbers (-24,24).



                    Visual check


    


     Plot y = x*(x+48) (red line),  y = -576 (green line)


Solved.



Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!

let two numbers be and
if they differing by we have
..........eq.1
Their product is
.......eq.2, substitute
Since ,


minimize the function by calculating '
'
Setting it to , we get


=>
then
..........eq.1

so, your numbers are and



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