Consider the two functions $y = x ^ { 2 } + a x + a$ and $y = x + 1$.
(1) The number of points at which the graphs of the two functions meet depends on the relationship of $a$ with the numbers $\mathbf { Q }$ and $\mathbf { R }$ in the following way: (For $\mathbf { N } \sim \mathbf { P }$ choose which of (0) $\sim$ (2) gives the correct condition for the question.)
(i) The condition under which the graphs of the two functions intersect at two different points is $\mathbf { N }$.
(ii) The condition under which the graphs of the two functions are tangent at a point is $\mathbf{O}$.
(iii) The condition under which the graph of $y = x ^ { 2 } + a x + a$ is always above the graph of $y = x + 1$ is $\mathbf { P }$.
$$\begin{aligned} & \text { (0) } \mathrm { Q } < a < \mathrm { R } \\ & \text { (1) } a = \mathrm { Q } \text { or } a = \mathrm { R } \\ & \text { (2) } a < \mathrm { Q } \text { or } \mathrm { R } < a \end{aligned}$$
(2) Let us consider the case where the value of $a$ satisfies P. Let $g ( x )$ be the difference between the values of the two functions, so $g ( x ) = x ^ { 2 } + a x + a - ( x + 1 )$, and let $m$ be the minimum value of $g ( x )$. Then
$$m = - \frac { \mathbf { S } } { \mathbf { T } } \left( a ^ { 2 } - \mathbf { U } a + \mathbf { U } \right)$$
Hence $m$ takes the maximum at $a = \mathbf { W }$ and its value there is $m = \mathbf { W }$.