To determine which function has the greater maximum value, we need to analyze both functions, [tex]\( f(x) = -3x^2 + 6x + 4 \)[/tex] and [tex]\( g(x) \)[/tex].
### Step 1: Find the Maximum Value of [tex]\( f(x) \)[/tex]
The function [tex]\( f(x) = -3x^2 + 6x + 4 \)[/tex] is a quadratic function of the form [tex]\( ax^2 + bx + c \)[/tex], where [tex]\( a = -3 \)[/tex], [tex]\( b = 6 \)[/tex], and [tex]\( c = 4 \)[/tex]. For a quadratic function [tex]\( ax^2 + bx + c \)[/tex] with [tex]\( a < 0 \)[/tex], the maximum value occurs at the vertex, which can be found using the formula [tex]\( x = -\frac{b}{2a} \)[/tex].
1. Calculate the x-coordinate of the vertex:
[tex]\[
x = -\frac{b}{2a} = -\frac{6}{2 \cdot (-3)} = -\frac{6}{-6} = 1
\][/tex]
2. Substitute [tex]\( x = 1 \)[/tex] back into the function to find the maximum value of [tex]\( f(x) \)[/tex]:
[tex]\[
f(1) = -3(1)^2 + 6(1) + 4 = -3 + 6 + 4 = 7
\][/tex]
So, the maximum value of [tex]\( f(x) \)[/tex] is 7.
### Step 2: Compare with the Maximum Value of [tex]\( g(x) \)[/tex]
We need to compare this maximum value with the maximum value of [tex]\( g(x) \)[/tex]. From the given information, we understand that the maximum value of [tex]\( g(x) \)[/tex] is 10.
### Step 3: Determine Which Function Has the Greater Maximum Value
We have:
- The maximum value of [tex]\( f(x) \)[/tex] is 7.
- The maximum value of [tex]\( g(x) \)[/tex] is 10.
Since 10 (the maximum value of [tex]\( g(x) \)[/tex]) is greater than 7 (the maximum value of [tex]\( f(x) \)[/tex]), we conclude that the function [tex]\( g(x) \)[/tex] has the greater maximum value.
The correct answer is:
C. [tex]\( g(x) \)[/tex]
