Find constants a, b, and c so that the graph of the function $f ( x ) = a x ^ { 2 } + b x + c$ has a relative maximum at (7, 34) and crosses the y-axis at (0, 6).
A) $a = \frac { 4 } { 7 }$ b=8,c=6
B) $a = - \frac { 4 } { 7 }$ , b=-8, c=-6
C) $a = \frac { 4 } { 7 }$ , b=-8, c=6
D) $a = - \frac { 4 } { 7 }$ , b=8, c= 6

Question

Quizplus · Accepted Answer

For a quadratic function $f(x) = ax^2 + bx + c$ to have a relative maximum at (7, 34), its derivative $f'(x) = 2ax + b$ must equal 0 when $x = 7$, because the slope of the tangent at a maximum point is 0. Substituting $x = 7$ into $f'(x) = 0$ gives $14a + b = 0$. The function crosses the y-axis at (0, 6), which means $f(0) = c = 6$. Substituting the point (7, 34) into the original function gives $49a + 7b + 6 = 34$. Solving these equations simultaneously, we find $a = -\frac{4}{7}$, $b = 8$, and $c = 6$, which corresponds to choice D.

Find Constants A, B, and C So That the Graph f(x)=ax2+bx+cf ( x ) = a x ^ { 2 } + b x + cf(x)=ax2+bx+c

Find Constants A, B, and C So That the Graph $f ( x ) = a x ^ { 2 } + b x + c$