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 $\mathbf { Z }$.
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 $\mathbf { Z }$.