Question

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

Answer 1

`f(x)` is convex at `x=1`.

Explanation:

`f(x)` is concave upward (convex) at a point `x_0` if `f''(x_0) > 0` and concave downward (concave) at a point `x_0` if `f''(x_0) < 0`.

In this case, we have

`f''(x) = d/dxf'(x)`

`=d/dx(d/dxxln(x)-x)`

`=d/dxln(x)`

`=1/x`

Then, at `x=1`:

`f'''(1) = 1/1 = 1 > 0`

Thus `f(x)` is convex at `x=1`.