Use the quadratic function to determine if the function has a maximum or minimum value and then find this maximum or minimum value.
-$$f ( x ) = - x ^ { 2 } - \frac { 4 } { 5 } x + 6$$
A) minimum at $\left( - \frac { 2 } { 5 } , - \frac { 121 } { 25 } \right)$
B) minimum at $\left( - \frac { 1 } { 10 } , - \frac { 73 } { 3 } \right)$
C) maximum at $\left( \frac { 1 } { 10 } , - 121 \right)$
D) maximum at $\left( - \frac { 2 } { 5 } , \frac { 154 } { 25 } \right)$

Question

Quizplus · Accepted Answer

The quadratic function is in the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. For the given function $f(x) = -x^2 - \frac{4}{5}x + 6$, the coefficient of $x^2$ is negative ($a = -1$), indicating that the parabola opens downwards and thus has a maximum value. To find the x-coordinate of the vertex (which gives the maximum or minimum point), use the formula $-\frac{b}{2a}$. Here, $a = -1$ and $b = -\frac{4}{5}$, so $-\frac{b}{2a} = -\frac{-\frac{4}{5}}{2(-1)} = -\frac{2}{5}$. Substituting $-\frac{2}{5}$ back into the original equation gives the y-coordinate of the vertex, which is the maximum value of the function. After calculation, the maximum value is found at $\left( -\frac{2}{5}, \frac{154}{25} \right)$.

Use the Quadratic Function to Determine If the Function Has f(x)=−x2−45x+6f ( x ) = - x ^ { 2 } - \frac { 4 } { 5 } x + 6f(x)=−x2−54​x+6

Use the Quadratic Function to Determine If the Function Has $f ( x ) = - x ^ { 2 } - \frac { 4 } { 5 } x + 6$