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As low as possible (Posted on 2013-04-12) Difficulty: 2 of 5
What is the lowest value of a product of two positive numbers a and b, given that a+b=a*b?

No Solution Yet Submitted by Ady TZIDON    
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Solution re(2): Analytical solution (spoiler) | Comment 5 of 6 |
(In reply to re: Analytical solution (spoiler) by Charlie)

Oops. sorry, misread it.  Let's try again:


(1) a + b = ab

(2) solving for b, we get b = a/(a-1)

    Then ab = a*a/(a-1)

(3) Let c = (a+1)
    
    Then ab = (c+1)*(c+1)/c = c + 2 + 1/c
    
    Derivative (with respect to c) = 1 - 1/(c*c)
    Setting this equal to 0 and solving gives c = 1
    
    So, local minimum occurs when ab = 1 + 2 + 1/1 = 4

  Posted by Steve Herman on 2013-04-12 14:11:03
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