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Logconcave 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

Curvature as a function of x

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