22. The function $f(x)=\frac{2}{3} x^{3}+\frac{1}{2}(a-1) x^{2}+a x+1$ is an increasing function on the interval $(1,2)$. Find the range of real numbers for $a$.


22. Since $f(x)$ is an increasing function on $(1,2)$, it follows that $f^{\prime}(x) \geqslant 0$ holds for all $x \in (1,2)$. The problem is thus transformed into finding the range of real numbers $a$ such that $f^{\prime}(x)=2 x^{2}+(a-1) x+a \geqslant 0$ for all $x \in (1,2)$.

By plotting the graph of $f^{\prime}(x)$ and using the axis of symmetry as a reference, we can discuss the following three cases. (See Figure 2-10) Using the graph and the monotonicity of the function, we get
$$
\left\{\begin{array} { l } 
{ \frac { 1 - a } { 4 } \leqslant 1 , } \\
{ f ^ { \prime } ( 1 ) \geqslant 0 }
\end{array} \text { or } \quad \left\{\begin{array} { l } 
{ 1 < \frac { 1 - a } { 4 } < 2 , } \\
{ \Delta = ( a - 1 ) ^ { 2 } - 8 a \leqslant 0 }
\end{array} \text { or } \quad \left\{\begin{array}{l}
\frac{1-a}{4} \geqslant 2, \\
f^{\prime}(2) \geqslant 0,
\end{array}\right.\right.\right.
$$

Solving these, we find $a \geqslant-\frac{1}{2}$, so the range of $a$ is $\left[-\frac{1}{2},+\infty\right)$.