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Math Help - Optimization Problem

  1. #1
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    Apr 2010
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    Optimization Problem

    been having a little trouble with this one:
    The sum of two positive numbers is 8. Show that the square of one plus the cube of the other is at least 44.
    Any help is appreciated
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  2. #2
    Senior Member
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    Jan 2010
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    Let a+b=8 with a and b positive. Then 0<a<8 and b=8-a.

    Without a loss of generality, we may choose to square b and cube a, so:

    a^3+b^2

    = a^3+(8-a)^2

    = a^3 + 64 - 16a + a^2

    = a^3+a^2-16a+64

    We want to find the minimum of this expression for 0<a<8.

    \frac{d}{da} (a^3+a^2-16a+64) = 0

    \implies 3a^2+2a-16=0

    \implies (3a+8)(a-2)=0

    This gives 2 critical points, but only one is in the region 0<a<8. So the minimum occurs at a=2. (You should probably show that it is in fact a minimum and not a maximum.)

    Now, you need to just evaluate the expression at a=2 to determine its minimum value.
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