639. Investigate the function $y=x^{3}-3 x^{2}-9 x+11$ for maximum, minimum, and point of inflection.
Solution. Let's find the first and second derivatives:

$$
f^{\prime}(x)=3 x^{2}-6 x-9=3\left(x^{2}-2 x-3\right) ; \quad f^{\prime \prime}(x)=6 x-6
$$

Set the first derivative to zero. From the equation $x^{2}-2 x-3=0$ we find $x_{1}=-1, x_{2}=3$. Substituting these values into the second derivative, we get $f^{\prime \prime}(-1)=-120$. Therefore, at $x=-1$ the function has a maximum, and at $x=3$ - a minimum.

Let's find the ordinates of the points of maximum and minimum: $f(-1)=-1-3+$ $+9+11=16, f(3)=-16$. Thus, $A(-1 ; 16)$ is the point of maximum, $B(3,-16)$ - the point of minimum.

Set the second derivative to zero. We have $f^{\prime \prime}(x)=6 x-6=0$, from which $x=1$. Determine the sign of $f^{\prime \prime}(x)$ in the intervals $(-\infty, 1)$ and $(1, \infty)$ : $f^{\prime \prime}(0)=-60$. Therefore, at $x=1$ the graph has a point of inflection. Let's find the ordinate of the point of inflection: $f(1)=1-3-9+11=$ $=0$. Thus, $C(1 ; 0)$ is the point of inflection of the given curve.

640-643. Investigate the concavity, convexity, and points of inflection of the functions: