Anonymous
Anonymous asked in Science & MathematicsMathematics · 1 month ago

find the critical numbers of f(x)=(x^2)/(x-1)?

the answer says its only x=0,2, why isn't 1 part of the answer since f' DNE at x=1

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  • TomV
    Lv 7
    1 month ago
    Favorite Answer

    (1, f(1)) is not a critical point because the definition of a critical point requires that the point be within the domain of the function.

    x = 1 is not in the domain of f(x) = x²/(x-1). Therefore f(1) is undefined and cannot be a critical point of f(x).

  • 1 month ago

    The technical definition of a critical point is where f '(x) = 0

    so, we could have local minimum, maximum or even points of inflection.

    The rational function does not exist for x = 1, as you state. In terms of analysing and sketching the function, x = 1 can be thought of as a critical value as x = 1 is a vertical asymptote.

    Using the quotient rule we have that:

    f '(x) = [2x(x - 1) - (1)(x²)]/(x - 1)²

    so, f '(x) = x(x - 2)/(x - 1)²

    Hence, f '(x) = 0 when x = 0 and when x = 2...which are critical points.

    It can be shown that (0, 0) is a local maximum and (2, 4) is a local minimum.

    A sketch is below.

    :)>

      

    Attachment image
  • Philip
    Lv 6
    1 month ago

    f(x) = (x^2)/(x-1) = (x^2)(x-1)^(-1);

    f'(x) = 2x(x-1)^(-1) -(x^2)(x-1)^(-2). (1/x)f'(x)(x-1)^2 = 2(x-1) -x = x-2, ie., 

    f'(x) = x(x-2)/(x-1)^2.;

    By definition, critical numbers of f(x) occur where f'(x) = 0. Not where f'(x) does

    not exist.

  • 1 month ago

    f(x) = x²/(x - 1)

    f'(x) = (d/dx)[x²/(x - 1)]

    f'(x) = [(x - 1)(x²)' - x²(x - 1)'] / (x - 1)²

    f'(x) = [(x - 1)(2x) - x²(1)] / (x - 1)²

    f'(x) = [2x² - 2x - x²] / (x - 1)²

    f'(x) = (x² - 2x) / (x - 1)²

    f'(x) = x(x - 2) / (x - 1)²

    When f'(x) = 0:

    x = 0 or x = 2

    Hence, f(x) has critical points at x = 0, 2

    There is NO critical point at x = 1.

    Actually, f(x) = 1/0 at x = 1

    In other words, f(x) is undefined at x = 1

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  • 1 month ago

    In a normal world x = 1 should be an answer as per definition of a critical point in calculus

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