Question: What is the concavity of (x)/(x-1)?
Write down the expression as
y = x/(x-1)
y is the curve
y', or dy/dx is the slope of the curve
y'', or d^2y/dx^2 is a measure of the concavity of the curve.
y'' > 0 shows that the curve is concave upwards.
y'' < 0 shows that the curve is concave downwards.
We have y = x/(x-1), then
y' = -1/(x-1)^2
y'' = 2/(x-1)^3
Now y'' > 0 for (x-1) > 0, i.e. x > 1
And y'' < 0 for (x-1) < 0, i.e. x < 1
Answer: y(x) is concave up for x > 1, and concave down for x < 1