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1. 
   neela Teacher
   Graduate School

   eNotes Editor

   The difference of the two numbers is 10, Therefore, the two numbers are like : x and x-10 or x and x+10. But without loss of generality we take x and x-10 to be the numbers whose doffrence is equal to 10.

   Therefore, thier product , P(x)= x(x-10)

   P(x) is minimum for x=a if P'(a) = 0 and p''(a) is positive.

   P'(x) = {x(x-10)}' = 2x-10. Equating 2x-10 to zero, we get x=5.

   P''(a) = At x=5, (2x-10)'  is 2 which is positive.

   Therefore, x=5 and x-10 = -5 . So 5 and 5-10 are the numbers having a diffrence of 10, with the least product 5*-5=-5^2=-25.

   Thefore, the minimum product of the two numbers whose dorence is 10 is -(10/2)^2 = -10^2/4 = -25

   Test: for any positive h,  (5+h)(-5+h) =h^2-25 >-25, as h^2 is positive.

   For any positive h, (5-h)(-5-h) =(-h)^2 -25 >-25, as (-h)^2=h^2 a positive quantity.

   Generalisation:

   If the two numbers have a given difference d, then  the two numbers for which the product is least  are -d/2 and -d/2 and the mimimum product is -(d/2)^2 = -d^2/4

   Hope this helps.

    

    

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   Posted by neela on Wednesday November 4, 2009 at 9:20 PM
   
   

2. 
   subbundd Teacher
   High School - 12th Grade

   We can answer by simple application of Differentiation.

   For either min or max the common condition is Ist derivative zero and for max  The IInd derivatime must be negative and for min the IINd derivative must be +ve.

   Let x be one number and the Iind number is x-10.

   P=x(x-10) =x^2-10x

   dP/dX=2x-10.

   d^2P/dx^2 =2 (+ve). Therefor minimum value at dP/dx=0

   2x-10=0 ie x=5.

   Therefore the numbers are 5 and -5.

   Min prodcuct = 58-5 =-25.

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   Posted by subbundd on Tuesday November 10, 2009 at 7:06 AM