이계도함수와 오목·볼록

Inflection points occur where $f''(x)$ changes sign. Since $f(x)$ is a polynomial, its first and second derivatives exist everywhere, and are themselves polynomials. In particular,

$$\begin{aligned}f(x)&=3x^5-5x^4+13x\\ f'(x)&=15x^4-20x^3+13\\ f''(x)&=60x^3-60x^2=60x^2(x-1) \end{aligned}$$

The second derivative is negative for $x<1$ and positive for $x>1$. Thus the concavity changes between concave up and concave down at $x=1,\ y=11$.

This is the only inflection point. It is true that $f''(0)=0$, but for values of $x$ both a little larger than and a little smaller than 0, $f''(x)<0$, so the concavity does not change at $x=0$.