Position of a curve relative to its tangent

1. $x\cos x=x-\frac{x^3}2+o(x^3)$. The tangent is $y=x$, and $f(x)-x\sim-\frac{x^3}2$ changes sign at $0$: the curve crosses its tangent (inflection point).

2. $\ln(1+x)=x-\frac{x^2}2+o(x^2)$. The tangent is $y=x$ and $g(x)-x\sim-\frac{x^2}2<0$: the curve lies below its tangent on both sides (locally concave).

3. $e^x\sin x=x+x^2+\frac{x^3}3+o(x^3)$. The tangent is $y=x$ and $h(x)-x\sim x^2>0$: the curve lies above its tangent (locally convex).

Rule. If the first non-zero term after the linear part is $c\,x^k$: for $k$ even, the sign of $c$ gives the side (above if $c>0$); for $k$ odd, the curve crosses the tangent.

Remark. Note that $x\cos x$ has $f''(0)=0$: this is the signature of an inflection point.