Question

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

Answer 1

Since `f''(-1)=2>0`, the function is convex (commonly called concave up) at `x=-1`.

Explanation:

The convexity and concavity of a function can be determined through its second derivative. At `x=a`, a function `f` is:

• convex (commonly known as concave up) if `f''(a)>0`
• concave (commonly known as concave down) if `f''(a)<0`

Find the function's second derivative by rewriting with a negative power then using the power rule:

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

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

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

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

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

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

The value of the second derivative at `x=-1` is:

`f''(-1)=-2/(-1)^3=-2/(-1)=2`

Since `f''(-1)=2>0`, the function is convex (commonly called concave up) at `x=-1`.