Question

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

Answer 1

`"concave at "x=1`

Explanation:

`"to determine if a function is concave/convex at f(a)"`
`"we require to evaluate "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)=x-1/x=x-x^-1`

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

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

`rArrf''(1)=-2<0`

`rArrf(x)" is concave at x = 1"`
graph{x-1/x [-10, 10, -5, 5]}