To analyze the quadratic function [tex]\( g(x) = 3x^2 + 12x + 16 \)[/tex], we will determine whether it has a minimum or maximum value, the value itself, and where it occurs. Let's go through the steps:
### Step 1: Determine if the function has a minimum or maximum value
The function [tex]\( g(x) = 3x^2 + 12x + 16 \)[/tex] is a quadratic function of the form [tex]\( ax^2 + bx + c \)[/tex], where [tex]\( a = 3 \)[/tex], [tex]\( b = 12 \)[/tex], and [tex]\( c = 16 \)[/tex]. For quadratic functions, the coefficient [tex]\( a \)[/tex] determines the shape of the parabola:
- If [tex]\( a > 0 \)[/tex], the parabola opens upward and has a minimum value.
- If [tex]\( a < 0 \)[/tex], the parabola opens downward and has a maximum value.
Since [tex]\( a = 3 \)[/tex] (which is greater than 0), the parabola opens upward, indicating that the function has a minimum value.
### Step 2: Find the x-coordinate where the minimum value occurs
The vertex of the parabola, which gives the minimum or maximum value, occurs at the x-coordinate given by the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Substituting the given values [tex]\( b = 12 \)[/tex] and [tex]\( a = 3 \)[/tex]:
[tex]\[ x = -\frac{12}{2 \times 3} = -\frac{12}{6} = -2 \][/tex]
So, the minimum value occurs at [tex]\( x = -2 \)[/tex].
### Step 3: Find the minimum value of the function
To find the minimum value of the function, substitute [tex]\( x = -2 \)[/tex] back into the quadratic function [tex]\( g(x) \)[/tex]:
[tex]\[ g(-2) = 3(-2)^2 + 12(-2) + 16 \][/tex]
Calculate each term step-by-step:
[tex]\[ 3(-2)^2 = 3 \times 4 = 12 \][/tex]
[tex]\[ 12(-2) = -24 \][/tex]
[tex]\[ 16 = 16 \][/tex]
Now, sum these values:
[tex]\[ g(-2) = 12 - 24 + 16 = 4 \][/tex]
Thus, the minimum value of the function is 4.
### Final results
- The function has a minimum value.
- The minimum value of the function is [tex]\( \boxed{4} \)[/tex].
- The minimum value occurs at [tex]\( x = \boxed{-2} \)[/tex].
