find 2 numbers whose sum is 54 and whose product is a maximum (using calculus)?

Obviously the 2 numbers are 27 and 27 but how do I prove that using calc

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  • 3 weeks ago

    Let our two number be a and b such that a = 27+x and b = 27-x, so a+b=54 and ab = 27² - x².

    The maximum value occurs when x² = 0, so when x = 0 a = b = 27.

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  • 3 weeks ago

    You have a number x 

    You want to maximize x(54-x)

    create a function y = 54x - x^2

    dy/dx = 54 - 2x

    x = 27 is where dy/dx = 0, from there determine whether it is a maximum or a minimum.

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  • 3 weeks ago

    Using ax²+bx+c=0 solve x₁ and x₂

    now x₁+x₂=-(b/a) and x₁.x₂=(c/a)

    so

    S=x₁+x₂ and P=x₁.x₂

    λ²-Sλ+P=0

    when derive by λ:

    2λ-S=0 (maximum or minimum) λ=S/2

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