perplexus.info :: Numbers : As low as possible

Home > Numbers

As low as possible (Posted on 2013-04-12)

What is the lowest value of a product of two positive numbers a and b, given that a+b=a*b?

See The Solution Submitted by Ady TZIDON

Rating: 2.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)

re(2): Analytical solution (spoiler)

| Comment 5 of 6 |

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

Please log in:

Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (3)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:


perplexus dot info