Determine whether the given quadratic function has a minimum value or maximum value. Then find the
coordinates of the minimum or maximum point.
-$$f ( x ) = - 3 x ^ { 2 } + 9 x$$
A) maximum; $\left( \frac { 3 } { 2 } , \frac { 27 } { 4 } \right)$
B) minimum; $\left( \frac { 3 } { 2 } , \frac { 27 } { 4 } \right)$
C) minimum; $\left( - \frac { 3 } { 2 } , - \frac { 27 } { 4 } \right)$
D) maximum; $\left( - \frac { 3 } { 2 } , - \frac { 27 } { 4 } \right)$

Question

Quizplus · Accepted Answer

The coefficient of $x^2$ in the quadratic function $f(x) = -3x^2 + 9x$ is negative, indicating the parabola opens downwards and thus has a maximum value. The vertex (maximum point) can be found using the formula $x = -\frac{b}{2a}$, where $a = -3$ and $b = 9$, giving $x = \frac{9}{2(-3)} = \frac{3}{2}$. Substituting $x = \frac{3}{2}$ into the function gives the y-coordinate of the vertex as $f\left(\frac{3}{2}\right) = -3\left(\frac{3}{2}\right)^2 + 9\left(\frac{3}{2}\right) = -\frac{27}{4} + \frac{27}{2} = \frac{27}{4}$. Therefore, the maximum point is $\left(\frac{3}{2}, \frac{27}{4}\right)$.

Determine Whether the Given Quadratic Function Has a Minimum Value f(x)=−3x2+9xf ( x ) = - 3 x ^ { 2 } + 9 xf(x)=−3x2+9x

Determine Whether the Given Quadratic Function Has a Minimum Value $f ( x ) = - 3 x ^ { 2 } + 9 x$