Convexity when First Derivative is a Constant

Question

I was trying to get the convexity of f(x) = x but then the first derivative was just a constant one and if I get the second derivative it would just give me a zero, does this mean that it's both concave and convex?

Answer 1

Yes, any linear function $f(x)=ax+b$ is both concave and convex. And these are the only functions with this property.

Answer 2

Yes, it is both concave and convex. You don't need derivatives for this. Just use the definitions of concave function and of convex function.