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 $\square \mathbf { J }$. Also, $m$ is minimized at $a = \mathbf { K }$ and the value of $m$ then is $\mathbf { L M }$.