Use the quadratic function to determine if the function has a maximum or minimum value and then find this maximum or minimum value.
-$$f ( x ) = - 3 x ^ { 2 } + 36 x - 102$$
A) minimum at $( - 6,0 )$
B) minimum at $( 0 , - 6 )$
C) maximum: $( 6,6 )$
D) maximum at $( - 6 , - 6 )$

Question

Quizplus · Accepted Answer

The quadratic function is in the form $f(x) = ax^2 + bx + c$, where $a = -3$, $b = 36$, and $c = -102$. Since $a < 0$, the parabola opens downwards, indicating a maximum value. To find the x-coordinate of the vertex (which gives the maximum or minimum point), use the formula $x = -\frac{b}{2a}$. Substituting the given values, $x = -\frac{36}{2(-3)} = 6$. To find the corresponding y-coordinate (the maximum value), substitute $x = 6$ into the original equation: $f(6) = -3(6)^2 + 36(6) - 102 = -108 + 216 - 102 = 6$. Therefore, the function has a maximum value at $(6, 6)$.

Use the Quadratic Function to Determine If the Function Has f(x)=−3x2+36x−102f ( x ) = - 3 x ^ { 2 } + 36 x - 102f(x)=−3x2+36x−102

Use the Quadratic Function to Determine If the Function Has $f ( x ) = - 3 x ^ { 2 } + 36 x - 102$