If non-zero real numbers $b$ and $c$ are such that $\min f ( x ) > \max g ( x )$, where $f ( x ) = x ^ { 2 } + 2 b x + 2 c ^ { 2 }$ and $g ( x ) = - x ^ { 2 } - 2 c x + b ^ { 2 } , ( x \in R )$; then $\left| \frac { c } { b } \right|$ lies in the interval
(1) $( \sqrt { 2 } , \infty )$
(2) $\left[ \frac { 1 } { 2 } , \frac { 1 } { \sqrt { 2 } } \right)$
(3) $\left( 0 , \frac { 1 } { 2 } \right)$
(4) $\left[ \frac { 1 } { \sqrt { 2 } } , \sqrt { 2 } \right]$

Consider the two parabolas $$\begin{aligned} \ell : & & y = ax^2 + 2bx + c \\ m : & & y = (a+1)x^2 + 2(b+2)x + c + 3. \end{aligned}$$ Four points A, B, C and D are assumed to be in the relative positions shown in the figure. One of the two parabolas passes through the three points A, B and C, and the other one passes through the three points B, C and D.
(1) The parabola passing through the three points A, B and C is $\mathbf{A}$. Here, for $\mathbf{A}$ choose the correct answer from (0) or (1), just below. (0) parabola $\ell$ (1) parabola $m$
(2) Since both parabolas $\ell$ and $m$ pass through the two points B and C, the $x$-coordinates of B and C are the solutions of the quadratic equation $$x^2 + \mathbf{B}x + \mathbf{C} = 0.$$ Hence, the $x$-coordinate of point B is $\mathbf{DE}$, and the $x$-coordinate of point C is $\mathbf{FG}$.
(3) In particular, we are to find the values of $a$, $b$ and $c$ when $\mathrm{AB} = \mathrm{BC}$ and $\mathrm{CO} = \mathrm{OD}$.
Since the two points C and D are symmetric with respect to the $y$-axis, we have $b = \mathbf{H}$. On the other hand, since $\mathrm{AB} = \mathrm{BC}$, the straight line $x = \mathbf{IJ}$ is the axis of symmetry of $\mathbf{A}$. Hence we have $a = -\frac{\mathbf{K}}{\mathbf{L}}$. And we have $c = \frac{\mathbf{M}}{\mathbf{L}}$.

Consider the two quadratic functions
$$f ( x ) = - 2 x ^ { 2 } , \quad g ( x ) = x ^ { 2 } + a x + b$$
Function $g ( x )$ satisfies the following two conditions:
(i) the value of $g ( x )$ is minimized at $x = 3$;
(ii) $g ( 4 ) = f ( 4 )$.
(1) From condition (i) we see that $a = -$ A . Further, from condition (ii) we see that $b = - \mathbf { B C }$. Hence the minimum value of function $g ( x )$ is $- \mathbf { D E }$.
(2) Let us find the value of $x$ such that $f ( x ) = g ( x )$ and $x$ is not 4 . Since $x$ satisfies
$$x ^ { 2 } - \mathbf { F } x - \mathbf { G } \mathbf { G } = 0 \text {, }$$
we obtain $x = - \mathbf { H }$.
(3) The value of $f ( x ) - g ( x )$ on $- \mathrm { H } \leqq x \leqq 4$ is maximized at $x = \square$, and its maximum value is JK.

Consider the two quadratic functions
$$f ( x ) = - 2 x ^ { 2 } , \quad g ( x ) = x ^ { 2 } + a x + b$$
Function $g ( x )$ satisfies the following two conditions:
(i) the value of $g ( x )$ is minimized at $x = 3$;
(ii) $\quad g ( 4 ) = f ( 4 )$.
(1) From condition (i) we see that $a = -$ A . Further, from condition (ii) we see that $b = - \mathbf { B C }$. Hence the minimum value of function $g ( x )$ is $- \mathbf { D E }$.
(2) Let us find the value of $x$ such that $f ( x ) = g ( x )$ and $x$ is not 4 . Since $x$ satisfies
$$x ^ { 2 } - \mathbf { F } x - \mathbf { G } \mathbf { G } = 0 \text {, }$$
we obtain $x = - \mathbf { H }$.
(3) The value of $f ( x ) - g ( x )$ on $- \mathrm { H } \leqq x \leqq 4$ is maximized at $x = \square$, and its maximum value is JK.

A family of quadratic curves is given by
$$y _ { k } = 2 \left( x - \frac { k } { 2 } \right) ^ { 2 } + \frac { k ^ { 2 } } { 2 } + 4 k + 3$$
where $k$ is any real number and $y _ { k }$ is a function of $x$.
All these curves are sketched, and the point with the lowest $y$-coordinate among all the curves $y _ { k }$ is $( a , b )$.
Find the value of $a + b$

The parabolas $f ( x )$ and $g ( x )$ whose graphs are shown above intersect each other at their vertices.
Given this, what is the value of $\mathbf { g } ( \mathbf { 0 } )$?
A) 3
B) 4
C) 5
D) 6
E) 7