Consider the two functions

$$\begin{aligned}
& f ( x ) = x ^ { 2 } + 2 a x + 4 a - 3 \\
& g ( x ) = 2 x + 1
\end{aligned}$$

We are to find the condition on $a$ for which $f ( x ) \geqq g ( x )$ for all $x$ and also find the range of values of the minimum of $f ( x )$ under this condition.

We must find the condition under which

$$x ^ { 2 } + \mathbf { A } ( a - \mathbf { A } ) x + \mathbf { A C } a - \mathbf { A D } \geq 0$$

for all $x$.

For each of $\mathbf { E } \sim \mathbf{ H }$ in the following questions, choose the correct answer from among (0) $\sim$ (7) below each question.

(1) The required condition is that $a$ satisfy the quadratic inequality $\mathbf{E}$. Hence $a$ is in the range $\mathbf { F }$.\\
(0) $a ^ { 2 } - 5 a + 4 \geqq 0$\\
(1) $a ^ { 2 } - 6 a + 5 \geqq 0$\\
(2) $a ^ { 2 } - 5 a + 4 \leqq 0$\\
(3) $a ^ { 2 } - 6 a + 5 \leqq 0$\\
(4) $a \leqq 1$ or $5 \leqq a$\\
(5) $1 \leqq a \leqq 5$\\
(6) $1 \leqq a \leqq 4$\\
(7) $a \leqq 1$ or $4 \leqq a$

(2) Let $m$ be the minimum value of $f ( x )$. Then, since $m = \mathbf { G }$, the range of values which $m$ can take under the condition in (1) is $\mathbf { H }$.\\
(0) $a ^ { 2 } + 4 a - 3$\\
(1) $4 a ^ { 2 } + 4 a - 3$\\
(2) $- a ^ { 2 } + 4 a - 3$\\
(3) $2 a ^ { 2 } - 4 a + 3$\\
(4) $- 5 \leqq m \leqq 1$\\
(5) $- 8 \leqq m \leqq 1$\\
(6) $- 8 \leqq m \leqq - 1$\\
(7) $- 5 \leqq m \leqq - 1$