Graph of $f(x)=|x|(\sin x)^{1/5}$ near $x=0$

Question

I would like to find the graph of $f(x)=|x|(\sin x)^{1/5}$ near x=0. How can I deduce the right inflection point without calculating the second derivative?

A: B: C:

Answer 1

You need an odd function approximating $x^{6/5}$ for small $x>0$, with an even derivative approximating $\frac65 x^{1/5}$ for such $x$. The former fact excludes $B$, because the exponent $>1$; the latter fact excludes $A$, whose gradient at the origin is nonzero.