Question

For what values of x is `f(x)=(-2x)/(x-1)` concave or convex?

Answer 1

Study the sign of the 2nd derivative.

For `x<1` the function is concave.
For `x>1` the function is convex.

Explanation:

You need to study curvature by finding the 2nd derivative.

`f(x)=-2x/(x-1)`

The 1st derivative:

`f'(x)=-2((x)'(x-1)-x(x-1)')/(x-1)^2`

`f'(x)=-2(1*(x-1)-x*1)/(x-1)^2`

`f'(x)=-2(x-1-x)/(x-1)^2`

`f'(x)=2*1/(x-1)^2`

The 2nd derivative:

`f''(x)=(2*(x-1)^-2)'`

`f''(x)=2((x-1)^-2)'`

`f''(x)=2*(-2)(x-1)^-3`

`f''(x)=-4/(x-1)^3`

Now the sign of `f''(x)` must be studied. The denominator is positive when:

`-(x-1)^3>0`
`(x-1)^3<0`
`(x-1)^3<0^3`
`x-1<0`
`x<1`

For `x<1` the function is concave.
For `x>1` the function is convex.

Note: the point `x=1` was excluded because the function `f(x)` can not be defined for `x=1`, since the denumirator would become 0.

Here is a graph so you can see with your eyes:

graph{(-2x)/(x-1) [-14.08, 17.95, -7.36, 8.66]}