Anonymous
Anonymous asked in Science & MathematicsMathematics Β· 2 years ago

Find the smallest real constant 𝑐 for all integers 𝑛 β‰₯ 10.?

Find the smallest real constant 𝑐 such that 3𝑛 2 + 101𝑛 ≀ 𝑐×(2𝑛 2 βˆ’ 19𝑛 + 20) for all integers 𝑛 β‰₯ 10.

Update:

Sorry it should be 3𝑛^2 + 101𝑛 ≀ 𝑐×(2𝑛^2 βˆ’ 19𝑛 + 20) for all integers 𝑛 β‰₯ 10.

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  • kb
    Lv 7
    2 years ago
    Favorite Answer

    Note that 3n^2 + 101n ≀ c(2n^2 βˆ’ 19n + 20) is equivalent to

    (2c - 3) n^2 - (19c + 101)n + 20c β‰₯ 0.

    Letting n = 10 yields

    100(2c - 3) - 10(19c + 101) + 20c β‰₯ 0

    <==> 30c β‰₯ 1310

    <==> c β‰₯ 131/3.

    We see that c = 131/3 suffices, by examining the plot (note that the curve is above the horizontal axis for all n β‰₯ 10):

    https://www.wolframalpha.com/input/?i=plot+y+%3D+(...

    I hope this helps!

  • 2 years ago

    3𝑛 2 + 101𝑛 ≀ 𝑐×(2𝑛 2 βˆ’ 19𝑛 + 20)

    ??

    can you rewrite this in understandable notation?

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