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
$$a = \mathbf { H } , \quad b = \mathbf { I } .$$

Q1 The quadratic function $f ( x ) = 2 x ^ { 2 } + a x - 1$ in $x$ satisfies
$$f ( - 1 ) \geqq - 3 , \quad f ( 2 ) \geqq 3 .$$
Let us consider the minimum value $m$ of $f ( x )$.
(1) $\quad m$ can be expressed in terms of $a$ as
$$m = -\frac{\mathbf{A}}{\mathbf{B}} a^2 - \mathbf{C}$$
(2) The range of the values of $a$ such that $f ( x )$ satisfies condition (1) is
$$\mathbf { D E } \leqq a \leqq \mathbf { F } .$$
(3) The value of $m$ is maximized when the axis of symmetry of the graph of $y = f ( x )$ is the straight line $x = \mathbf { G }$, and then the value of $m$ is $\mathbf { H I }$.
(4) The value of $m$ is minimized when the axis of symmetry of the graph of $y = f ( x )$ is the straight line $x = \mathbf { J K }$, and then the value of $m$ is $\mathbf { L M }$.

Q1 The quadratic function $f ( x ) = 2 x ^ { 2 } + a x - 1$ in $x$ satisfies
$$f ( - 1 ) \geqq - 3 , \quad f ( 2 ) \geqq 3 .$$
Let us consider the minimum value $m$ of $f ( x )$.
(1) $m$ can be expressed in terms of $a$ as
$$m = - \frac { \mathbf { A } } { \mathbf{B} } a ^ { 2 } - \mathbf { C }$$
(2) The range of the values of $a$ such that $f ( x )$ satisfies condition (1) is
$$\mathbf { D E } \leqq a \leqq \mathbf { F } .$$
(3) The value of $m$ is maximized when the axis of symmetry of the graph of $y = f ( x )$ is the straight line $x = \mathbf { G }$, and then the value of $m$ is $\mathbf { H I }$.
(4) The value of $m$ is minimized when the axis of symmetry of the graph of $y = f ( x )$ is the straight line $x = \mathbf { J K }$, and then the value of $m$ is $\mathbf { L M }$.

Let us consider the maximum value $M$ and the minimum value $m$ of the quadratic function
$$f ( x ) = x ^ { 2 } - 2 ( a + 1 ) x + 2 a ^ { 2 }$$
over $0 \leqq x \leqq 2$, where $a$ is a constant and $0 \leqq a \leqq 3$.
(1) The coordinates of the vertex of the graph of $y = f ( x )$ are
$$\left( a + \mathbf { A } , a ^ { 2 } - \mathbf { B } a - \mathbf { C } \right) .$$
(2) For $\mathbf { D } \sim$ H in the following sentences, choose the correct answers from among choices (0) $\sim$ (9) below.
Let us find the maximum value $M$ and the minimum value $m$ according to the position of the axis of symmetry. We have that if $0 \leqq a < \mathbf { D }$, then
$$M = \mathbf { E } , \quad m = \mathbf { F } ;$$
if $\mathrm { D } \leqq a \leqq 3$, then
$$M = \mathbf { G } , \quad m = \mathbf { H } .$$
(0) 0
(1) 1
(2) 2
(3) 3
(4) $a ^ { 2 } - 2 a$
(5) $a ^ { 2 } - 2 a - 1$ (6) $2 a ^ { 2 }$ (7) $2 a ^ { 2 } - 2 a - 1$ (8) $2 a ^ { 2 } - 4 a$ (9) $2 a ^ { 2 } - 6 a + 3$
(3) Thus, $m$ is maximized at $a = \square$ and the value of $m$ then is $\square \mathbf { J }$. Also, $m$ is minimized at $a = \mathbf { K }$ and the value of $m$ then is $\mathbf { L M }$.

Let us consider the maximum value $M$ and the minimum value $m$ of the quadratic function
$$f ( x ) = x ^ { 2 } - 2 ( a + 1 ) x + 2 a ^ { 2 }$$
over $0 \leqq x \leqq 2$, where $a$ is a constant and $0 \leqq a \leqq 3$.
(1) The coordinates of the vertex of the graph of $y = f ( x )$ are
$$\left( a + \mathbf { A } , a ^ { 2 } - \mathbf { B } a - \mathbf { C } \right) .$$
(2) For $\mathbf { D } \sim$ H in the following sentences, choose the correct answers from among choices (0) $\sim$ (9) below.
Let us find the maximum value $M$ and the minimum value $m$ according to the position of the axis of symmetry. We have that if $0 \leqq a < \mathbf { D }$, then
$$M = \mathbf { E } , \quad m = \mathbf { F } ;$$
if $\mathrm { D } \leqq a \leqq 3$, then
$$M = \mathbf { G } , \quad m = \mathbf { H } .$$
(0) 0
(1) 1
(2) 2
(3) 3
(4) $a ^ { 2 } - 2 a$
(5) $a ^ { 2 } - 2 a - 1$ (6) $2 a ^ { 2 }$ (7) $2 a ^ { 2 } - 2 a - 1$ (8) $2 a ^ { 2 } - 4 a$ (9) $2 a ^ { 2 } - 6 a + 3$
(3) Thus, $m$ is maximized at $a = \square$ and the value of $m$ then is $\square$ J. Also, $m$ is minimized at $a = \mathbf { K }$ and the value of $m$ then is $\mathbf { L M }$.

The function $f ( x ) = x ^ { 2 } + a x + b$ satisfies the following two conditions:
(i) $\quad f ( 3 ) = 1$;
(ii) $13 \leqq f ( - 1 ) \leqq 25$.
We are to express the minimum value $m$ of $f ( x )$ in terms of $a$. In addition, we are to find the maximum and minimum values of $m$.
From condition (i), $a$ and $b$ satisfy
$$\mathbf { N } a + b + \mathbf { O } = 0 \text {. }$$
From this, $f ( x )$ can be expressed in terms of $a$ as
$$f ( x ) = x ^ { 2 } + a x - \mathbf { P } a - \mathbf { Q } .$$
Hence from condition (ii), $a$ satisfies
$$- \mathbf { R } \leqq a \leqq - \mathbf { S } .$$
On the other hand, $m$ can be expressed in terms of $a$ as
$$m = - \frac { 1 } { \mathbf { T } } ( a + \mathbf { U } ) ^ { 2 } + \mathbf { V }$$
Thus $m$ is maximized at $a = - \mathbf { W }$, and its maximum value is $\mathbf { X }$; it is minimized at $a = - \mathbf { Y }$, and its minimum value is $\mathbf { Z }$.

Let the quadratic function $f ( x ) = x ^ { 2 } + b x + c$, where $b , c$ are real numbers. Given that $f ( x - 2 ) = f ( - x - 2 )$ holds for all real numbers $x$, and when $- 3 \leq x \leq 1$, the maximum value of $f ( x )$ is 4 times its minimum value, what is the minimum value of $f ( x )$?
(1) 0
(2) $\frac { 5 } { 3 }$
(3) 3
(4) 4
(5) 6