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    Logconvex Function

    Result

    y = log(x) is concave down | on x<0 concave down | on x>0

    Plot

    | | concave up | concave down

    Critical points

    (no critical points)

    Derivative

    d/dx(log(x)) = 1/x

    Critical points of the derivative

    (no critical points)

    Second derivative

    d^2/dx^2(log(x)) = -1/x^2

    Domain

    {x element R : x>0} (all positive real numbers)

    Discontinuities

    x = 0 (infinite discontinuity) (assuming a function from reals to reals)

    Curvature

    abs(x)/(x^2 + 1)^(3/2)

    Osculating circle at x_0

    center | (2 x_0 + 1/x_0, -x_0^2 + log(x_0) - 1) radius | (x_0^2 + 1)^(3/2)/abs(x_0) equation | (x - 1/x_0 - 2 x_0)^2 + (1 + y - log(x_0) + x_0^2)^2 = (1 + x_0^2)^3/abs(x_0)^2

    Plot of osculating circles

    Plot of osculating circles

    Curvature as a function of x

    Curvature as a function of x

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