View Full Version : Maxima and minima
akotoh
Oct 5, 2009, 08:27 AM
Please help me with this one...
Find two positive numbers whose product is 64, and whose sum is a minimum. Thanks!
Perito
Oct 5, 2009, 08:56 AM
X = first number
Y = second number
XY = 64
X + Y = minimum.
Substituting
X + 64/X = minimum
To find the minimum (or maximum), differentiate with respect to X and set the result to zero.
\frac {d}{dX}(X + \frac {64}{X}) = 1 - \frac {64}{X^2} = 0
\frac {64}{X^2} = 1
64 = X^2
X = \pm\,8
Y = \frac {64}{X}
Therefore
Y = \pm \, 8
If you want the sum to be a minimum then you must select X=-8, Y=-8
p.s.
... But, since the problem specified positive numbers, you select
X=8 and Y=8 :o
ebaines
Oct 5, 2009, 08:56 AM
I suggest you write out all the ways you can multiply two positive numbers to get 64, and see which has the smallest sum:
1*64
2*32
4*16
etc.
EDIT - I assumed (perhaps incorrectly) that the OP meant two positive "integers," in which case this technique is adequate. I also assumed (again, I suspect incorrectly) that this was a question for someone in elementary school, and hence the use of calculus as Perito suggested would be beyond the level of understanding of the OP. However, I see from some earlier posts that the OP is indeed studying derivatives, so Perito's technique is appropriate.
Unknown008
Oct 5, 2009, 09:32 AM
Well, Perito you missed the part where it said that 'two positive numbers' :rolleyes:
Perito
Oct 5, 2009, 02:02 PM
Oops.