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    [FONT='Cambria','serif']There are 2 positive numbers who have a sum of 9. Determine the maximum value of the product of one number and the square of the other number.[/FONT]
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    Quote Originally Posted by someone21 View Post
    [FONT='Cambria','serif']There are 2 positive numbers who have a sum of 9. Determine the maximum value of the product of one number and the square of the other number.[/FONT]
    Let x, y denote the two positive numbers.

    Then you know:

    x+y=9~\implies~x = 9-y ...... [1]

    p = x \cdot y^2 ...... [2]

    Plug in the term of [1] into [2]:

    p(y) = (9-y) \cdot y^2 = -y^3+9y^2

    Calculate the first derivative and solve the equation p'(y) = 0 for y.

    I've got y = 0 or y = 6 and therefore the maximum value of the product is 108.
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  3. #3
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    Quote Originally Posted by someone21 View Post
    [FONT='Cambria','serif']There are 2 positive numbers who have a sum of 9. Determine the maximum value of the product of one number and the square of the other number.[/FONT]
    a+b = 9. This means ab^2 = (9 - b)b^2 = 9b^2 - b^3 = f(b)

    Maxima of f(b) is when f'(b) = 0 . So f'(b) = 18b - 3b^2 = 0 \Rightarrow b = 6. b = 0 is the minima.
    Thus a = 3 and the maximum product is 3.6^2 = 108
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  4. #4
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by someone21 View Post
    [FONT='Cambria','serif']There are 2 positive numbers who have a sum of 9. Determine the maximum value of the product of one number and the square of the other number.[/font]
    let x be one of the numbers. so the other can be represented as 9 - x

    Let P(x) be the function representing the product as stated WLOG: P(x) = x^2(9-x) = 9x^2 - x^3

    so, P'(x) = 18x - 3x^2 = 3x(6-x) and we want the maximum, thus we need x such that P'(x) = 0

    therefore, either x=0 or x=6 but note that x is a positive number, therefore we take x=6.

    -----------------
    optional: if P''(x_0) < 0, then we are sure that x_0 gives the maximum value for P(x)

    \Rightarrow P''(x) = 18 - 6x and at x=6, we have  P''(6) = 18 - 36 = -18 < 0.
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