Classifying a critical point when f″(0) = 0

Rule. If $f(x)=c\,x^m+o(x^m)$ with $c\ne0$, then near $0$ the function $f$ has the sign of $cx^m$.

1. $\cos x=1-\frac{x^2}2+\frac{x^4}{24}-\dots$, so $f_1(x)=\dfrac{x^4}{24}+o(x^4)$. The exponent $4$ is even and the coefficient is positive: $f_1\ge0$ near $0=f_1(0)$. Local minimum.

2. $\sin x=x-\frac{x^3}6+\frac{x^5}{120}-\dots$, so $f_2(x)=\dfrac{x^5}{120}+o(x^5)$. The exponent $5$ is odd: $f_2$ changes sign at $0$. No extremum (an inflection point).

3. $x\sin x=x^2-\frac{x^4}6+\dots$, so $f_3(x)=-\dfrac{x^4}6+o(x^4)$. The exponent is even and the coefficient is negative: local maximum.

Remark. The second derivative test ($f''(0)>0$ or $<0$) is silent in these three cases because $f''(0)=0$. Taylor expansions give the complete picture.