636. Investigate the convexity, concavity, and points of inflection of the curve $f(x)=x^{3}-3 x^{2}+5$.
Solution. $1^{0}$. Determine the first and second derivatives: $f^{\prime}(x)=$ $=3 x^{2}-6 x ; f^{\prime \prime}(x)=6 x-6=6(x-1)$.

$2^{0}$. From the equation $f^{\prime \prime}(x)=0$ we have $6(x-1)=0$, i.e., $x=1$ is a critical point of the second kind.

$3^{0}$. If $x<1$, then $f^{\prime \prime}(x)<0$ and in the interval $(-\infty, 1)$ the curve is convex; if $x>1$, then $f^{\prime \prime}(x)>0$ and in the interval $(1, \infty)$ the curve is concave, and $x=1$ is the abscissa of the inflection point.

$4^{0}$. At $x=1$ we get $f(1)=3$, i.e., $M(1 ; 3)$ is the inflection point.