Consider the quadratic function in $x$ $$y = - \frac { 1 } { 8 } x ^ { 2 } + a x + b .$$ When we denote the coordinates of the vertex of the graph of (1) by $( p , q )$, we have $$p = \mathbf { A } a , \quad q = \mathbf { B } a ^ { 2 } + b .$$ (1) When the vertex ( $p , q$ ) is on the straight line $x + y = 1 , a$ and $b$ satisfy $$b = \mathbf { C D } a ^ { 2 } - \mathbf { E E } a + \mathbf { F } ,$$ and so $8 a + b$ is maximized at $a = \mathbf { G }$, and the maximum value is $\mathbf { H }$. (2) When the graph of (1) is tangent to the $x$-axis, the range of values of $a + b$ is $$a + b \leqq \frac { \mathbf { I } } { \mathbf { J } }$$
Consider the quadratic function in $x$
$$y = - \frac { 1 } { 8 } x ^ { 2 } + a x + b .$$
When we denote the coordinates of the vertex of the graph of (1) by $( p , q )$, we have
$$p = \mathbf { A } a , \quad q = \mathbf { B } a ^ { 2 } + b .$$
(1) When the vertex ( $p , q$ ) is on the straight line $x + y = 1 , a$ and $b$ satisfy
$$b = \mathbf { C D } a ^ { 2 } - \mathbf { E E } a + \mathbf { F } ,$$
and so $8 a + b$ is maximized at $a = \mathbf { G }$, and the maximum value is $\mathbf { H }$.
(2) When the graph of (1) is tangent to the $x$-axis, the range of values of $a + b$ is
$$a + b \leqq \frac { \mathbf { I } } { \mathbf { J } }$$