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Given [(x-5)/(x-3)]^2 . Find any stationary points and any points of inflection. Also find any horizontal and
vertical asymptotes.

  • Math -

    Stationary points are points at which f'(x)=0 AND f"(x)=0.
    Vertical asymptote is where the denominator becomes zero.
    Horizontal asymptote is the limit when x->∞ or x->-∞, if the limit is defined.

  • Math -

    could you please solve the problem completely .. i am still confused.

  • Math -

    Denominator becomes zero when x=3, therefore vertical asymptote is a x=3.

    The limit of f(x)=[(x-5)/(x-3)]^2 as x->±∞ is 1, so the horizontal asymptote is at y=1.

    Since f'(5)=0 and f"(6)=0, there are no stationary points, just a minimum at x=5.
    There is a point of inflection at x=6 (where f"(6)=0).

    You will need to take the time to calculate f'(x) and f"(x) from f(x), and hence sketch the graph of f(x).

    A sketch of the graph can be seen here:

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