Help regarding a calculus problem?

excelandconquerexcelandconquer
in Science & Mathematics

Hi I need help with this problem

Find two positive numbers x and y whose sum is 8, such that the sum of x^3 and y2 is a minimum.

(i) Translate the first given condition into an equation relating the two numbers.

(ii) Use this condition to write the function you want to minimize in terms of x alone.

I dont understand what the first question is asking for, is the answer x^3+y^2 = 8 ? Please help I really need to know.

Comments

  • mohanrao-dmohanrao-d

    The first given condition is sum of two numbers is 8

    so x + y = 8, where x and y are positive numbers

    y = 8 - x

    second condition is x^3 + y^2 is minimum

    replacing y with 8-x

    x^3 + (8-x)^2 is minimum

    x^3 + x^2 - 16x + 64

    find the derivative

    3x^2 + 2x - 16

    in order to satisfy second condition, derivative should be zero.

    3x^2 + 2x - 16 = 0

    3x^2 -6x + 8x - 16 = 0

    3x(x - 2) + 8(x-2) = 0

    (x-2)(3x+8) = 0

    x = 2 (ignoring negative value)

    so y = 6

    so the two numbers are 2 and 6

  • cattbarfcattbarf

    Don't guess. Just read the problem. Let x and y be the two numbers.

    So x+y=8

    And, we want to minimize x^3+y^2

    Typically, d(x^3+y^2)/d(something) = 0

    So if we let y=8-x, then,

    x^3 +(8-x)^2 = SUM

    Then d(SUM)/dx = 3x^2 + 2x -16

    So at the minimum 3 x^2+ 2x - 16= 0

    (3x+8)(x-2)=0

    x=2 is only admissible answer

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