Question

Is `f(x)=1/(x-1)-2x` concave or convex at `x=0`?

Answer 1

`"concave at x = 0"`

Explanation:

To determine if f(x) is concave/convex at f( a) we require to find the value of f''( a)

`• "If " f''(a)> 0" then f(x) is convex at x = a"`

`• "If " f''(a)<0" then f(x) is concave at x = a"`

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

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

`rArrf''(x)=2(x-1)^-3=2/(x-1)^3`

`rArrf''(0)=2/(-1)^3=-2`

`"Since " f''(0)<0" then f(x) is concave at x = 0"`
graph{(1/(x-1))-2x [-10, 10, -5, 5]}