Question

Find the values of b such that the function has the given maximum value.

Answer 1

So, the equation has a leading term with a negative coefficient, and is a quadratic, meaning is a parabola opening downwards, so it goes up and up and up and then makes a U-turn then down down, so it has a maximum point, that is, at the vertex point

so.. where the dickens is the vertex at?
we're asked to make the vertex with a y-coordinate of 104, so the maximum value is 104

well
$\bf y = {{ a}}x^2{{ +b}}x{{ +c}}\qquad \left(-\cfrac{{{ b}}}{2{{ a}}}\quad ,\quad {{ c}}-\cfrac{{{ b}}^2}{4{{ a}}}\right)\\\\ -----------------------------\\\\ \textit{so we want }\implies {{ c}}-\cfrac{{{ b}}^2}{4{{ a}}}=104 \\\\ \textit{let's take a peek at your equation}\quad \begin{array}{ccccllll} f(x)=&-x^2&+bx&-17\\ f(x)=&-1x^2&+bx&-17\\ &\uparrow &\uparrow &\uparrow \\ &a&b&c \end{array} \\\\ thus \\\\ (-17)=\cfrac{b^2}{4(-1)}=104$

solve for "b"