To identify the vertex of the function [tex]\( R(x) = -x^2 + 6x \)[/tex], we can use the standard formula for the vertex of a quadratic function in the form [tex]\( ax^2 + bx + c \)[/tex].
The x-coordinate of the vertex of a parabola given by [tex]\( y = ax^2 + bx + c \)[/tex] can be found using:
[tex]\[ x = -\frac{b}{2a} \][/tex]
For the given function [tex]\( R(x) = -x^2 + 6x \)[/tex]:
- [tex]\( a = -1 \)[/tex]
- [tex]\( b = 6 \)[/tex]
Using the vertex formula:
[tex]\[ x = -\frac{6}{2(-1)} = \frac{6}{2} = 3 \][/tex]
Now, we find the y-coordinate by substituting [tex]\( x = 3 \)[/tex] back into the function [tex]\( R(x) \)[/tex]:
[tex]\[ R(3) = -(3)^2 + 6(3) = -9 + 18 = 9 \][/tex]
Therefore, the vertex of the revenue function [tex]\( R(x) \)[/tex] is at:
[tex]\[ (3, 9) \][/tex]
So, the vertex of [tex]\( R(x) \)[/tex] is at [tex]\( \boxed{(3, 9)} \)[/tex].
