Determine whether there is a maximum or minimum value for the given function, and find that value.
-$$f ( x ) = - x ^ { 2 } + 5 x - 3$$
A) Maximum: $\frac { 13 } { 4 }$
B) Minimum: $- 3$
C) Minimum: $\frac { 13 } { 4 }$
D) Maximum: $- \frac { 87 } { 4 }$

Question

Quizplus · Accepted Answer

The given function is a quadratic function in the form $f(x) = ax^2 + bx + c$, where $a = -1$, $b = 5$, and $c = -3$. Since $a < 0$, the parabola opens downwards, indicating a maximum value. To find the maximum value, use the vertex formula $x = -\frac{b}{2a}$ to find the x-coordinate of the vertex. Substituting $a = -1$ and $b = 5$ gives $x = -\frac{5}{2(-1)} = \frac{5}{2}$. Then, substitute $x = \frac{5}{2}$ into the original equation to find the y-coordinate (maximum value): $f(\frac{5}{2}) = -(\frac{5}{2})^2 + 5(\frac{5}{2}) - 3 = -\frac{25}{4} + \frac{25}{2} - 3 = \frac{25}{2} - \frac{25}{4} - \frac{12}{4} = \frac{13}{4}$.

Determine Whether There Is a Maximum or Minimum Value for the Given