Determine whether the given quadratic function has a minimum value or maximum value. Then find the coordinates of
the minimum or maximum point.
-$$f ( x ) = - 2 x ^ { 2 } + 6 x$$
A) minimum; $\left( - \frac { 3 } { 2 } , - \frac { 9 } { 2 } \right)$
B) minimum; $\left( \frac { 3 } { 2 } , \frac { 9 } { 2 } \right)$
C) maximum; $\left( - \frac { 3 } { 2 } , - \frac { 9 } { 2 } \right)$
D) maximum; $\left( \frac { 3 } { 2 } , \frac { 9 } { 2 } \right)$

Question

Quizplus · Accepted Answer

The coefficient of $x^2$ in the quadratic function $f(x) = -2x^2 + 6x$ is negative, indicating that the parabola opens downwards and thus has a maximum value. To find the x-coordinate of the vertex (maximum point), use the formula $-\frac{b}{2a}$, where $a = -2$ and $b = 6$. This gives $-\frac{6}{2(-2)} = \frac{3}{2}$. Substituting $x = \frac{3}{2}$ into the function gives $f\left(\frac{3}{2}\right) = -2\left(\frac{3}{2}\right)^2 + 6\left(\frac{3}{2}\right) = -2\left(\frac{9}{4}\right) + 9 = -\frac{9}{2} + 9 = \frac{9}{2}$. Therefore, the maximum point is $\left(\frac{3}{2}, \frac{9}{2}\right)$.

Determine Whether the Given Quadratic Function Has a Minimum Value f(x)=−2x2+6xf ( x ) = - 2 x ^ { 2 } + 6 xf(x)=−2x2+6x

Determine Whether the Given Quadratic Function Has a Minimum Value $f ( x ) = - 2 x ^ { 2 } + 6 x$