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24 March, 01:23

Find two positive numbers whose product is 81 and whose sum is a minimum.

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  1. 24 March, 01:34
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    Ab=81, a=81/b

    s=a+b using a from above in this we get:

    s (b) = 81/b + b

    s (b) = (81+b^2) / b

    ds/db = (2b*b-81-b^2) / b^2

    ds/db = (2b^2-81-b^2) / b^2

    ds/db = (b^2-81) / b^2

    d2s/db2 = (2b^3-2b^3+81) / b^4

    d2s/db2=81/b^4 since b is positive we know that the acceleration is positive so that when ds/db=0 it is a minimum for s (b)

    ds/db=0 only when b^2-81=0, b^2=81, b=9

    The two positive numbers are 9 and 9.
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