To determine which statement is true, we need to find the minimum values of the given quadratic functions [tex]\( f \)[/tex] and [tex]\( g(x) \)[/tex], and compare them.
### Step-by-Step Solution:
1. Function [tex]\( f \)[/tex] Analysis:
- Given: [tex]\( f \)[/tex] has a vertex at [tex]\( (3, 4) \)[/tex] and opens upward.
- The general form of a quadratic function is [tex]\( f(x) = a(x-h)^2 + k \)[/tex], where [tex]\((h, k)\)[/tex] is the vertex.
- Since [tex]\( f \)[/tex] opens upward, [tex]\( a > 0 \)[/tex].
- Therefore, the minimum value of [tex]\( f \)[/tex] is [tex]\( k \)[/tex].
- In this case, [tex]\( k = 4 \)[/tex].
- Hence, the minimum value of [tex]\( f \)[/tex] is [tex]\( 4 \)[/tex].
2. Function [tex]\( g(x) \)[/tex] Analysis:
- Given: [tex]\( g(x) = 2(x-4)^2 + 3 \)[/tex].
- The vertex form [tex]\( g(x) = a(x-h)^2 + k \)[/tex] tells us the vertex of [tex]\( g \)[/tex] is [tex]\((4, 3)\)[/tex].
- Since the parabola opens upward (as the coefficient of the squared term [tex]\( 2 \)[/tex] is positive), the minimum value of [tex]\( g \)[/tex] is the [tex]\( y \)[/tex]-coordinate of the vertex.
- Thus, the minimum value of [tex]\( g \)[/tex] is [tex]\( 3 \)[/tex].
3. Comparing the Minimum Values:
- The minimum value of [tex]\( f \)[/tex] is [tex]\( 4 \)[/tex].
- The minimum value of [tex]\( g \)[/tex] is [tex]\( 3 \)[/tex].
Clearly, the minimum value of [tex]\( f \)[/tex] is greater than the minimum value of [tex]\( g \)[/tex].
Given this analysis, we can conclude that:
A. The minimum value of [tex]\( f \)[/tex] is greater than the minimum value of [tex]\( g \)[/tex].
So the correct answer is:
```
1
```
which corresponds to statement:
A.
