Let $a$ and $b$ be real numbers where $0 < b < 7$. Let us consider the maximum value $M$ and the minimum value $m$ of the quadratic function $$f ( x ) = x ^ { 2 } - 6 x + a$$ over the interval $b \leqq x \leqq 7$. The function $f ( x )$ can be represented as $$f ( x ) = ( x - \mathbf { A } ) ^ { 2 } + a - \mathbf { B } .$$ (1) For each of $\mathbf { C }$ ~ $\mathbf { G }$ in the following statements, choose the correct answer from among (0) ~ (9) below. We are to find $M$ and $m$. There are two cases. (i) When $0 < b \leqq \mathbf { C }$, then $$M = \mathbf { D } , \quad m = \mathbf { E } .$$ (ii) When $\mathbf { C } < b < 7$, then $$M = \mathbf { F } , \quad m = \mathbf { G } .$$ (0) 0 (1) 1 (2) 2 (3) 3 (4) $a - 6$ (5) $a + 7$ (6) $a + 8$ (7) $a - 9$ (8) $b ^ { 2 } - 6 b + a$ (9) $b ^ { 2 } + 6 b + a$ (2) In the case that $M = 13$ and $m = 1$, we have $$a = \mathbf { H } , \quad b = \mathbf { I } .$$
Let $a$ and $b$ be real numbers where $0 < b < 7$. Let us consider the maximum value $M$ and the minimum value $m$ of the quadratic function
$$f ( x ) = x ^ { 2 } - 6 x + a$$
over the interval $b \leqq x \leqq 7$.
The function $f ( x )$ can be represented as
$$f ( x ) = ( x - \mathbf { A } ) ^ { 2 } + a - \mathbf { B } .$$
(1) For each of $\mathbf { C }$ ~ $\mathbf { G }$ in the following statements, choose the correct answer from among (0) ~ (9) below.
We are to find $M$ and $m$. There are two cases.\\
(i) When $0 < b \leqq \mathbf { C }$, then
$$M = \mathbf { D } , \quad m = \mathbf { E } .$$
(ii) When $\mathbf { C } < b < 7$, then
$$M = \mathbf { F } , \quad m = \mathbf { G } .$$
(0) 0\\
(1) 1\\
(2) 2\\
(3) 3\\
(4) $a - 6$\\
(5) $a + 7$\\
(6) $a + 8$\\
(7) $a - 9$\\
(8) $b ^ { 2 } - 6 b + a$\\
(9) $b ^ { 2 } + 6 b + a$
(2) In the case that $M = 13$ and $m = 1$, we have
$$a = \mathbf { H } , \quad b = \mathbf { I } .$$