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calculus

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Find constants a and b in the function f(x)=ax^b/(ln(x)) such that f(19)=1 and the function has a local minimum at x=19. What is a and b?

  • calculus -

    Just plug and chug:

    f = ax^b/lnx
    1 = a*19^b / ln 19

    a * 19^b = 2.944

    f' = (abx^(b-1) * lnx) - ax^(b-1))/ln^2(x)
    = [ax^(b-1) * (b*ln x-1)]/ln^2(x)

    to get f'=0, we need b*lnx = 1

    b*ln19 = 1
    b = 1/ln19

    a* 19^b = 2.944
    a*19^(1/ln19) = a * 19^.3396 = a*2.718 = 2.944
    a = 1.083

    So, if my math is right, f(x) = 1.083*x^.3396/lnx

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