Inflection point for function with fractional exponents

Because $f''(1)<0$, this point is a local maximum, not an inflection point, and the book is wrong.

$x=1$ is not an inflection point (as you can see from the graph below).

On the other hand you missed another inflection point $x=0$! Perhaps your book mistakenly wrote $x=1$ instead of $x=0$. The reason you missed $0$ is because the function is not differentiable at $0$ (and for such points, the condition of $f''(x)=0$ for inflection points obviously fails).