To determine whether the given quadratic function [tex]\( f(x) = 3x^2 + 3x - 9 \)[/tex] has a minimum value or maximum value and to find the coordinates of this point, we need to follow these steps:
1. Identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] from the standard quadratic form [tex]\( ax^2 + bx + c \)[/tex]. Here, we have:
[tex]\[
a = 3, \quad b = 3, \quad c = -9
\][/tex]
2. Determine if the quadratic function has a minimum or maximum value:
- If [tex]\( a > 0 \)[/tex], the parabola opens upwards, indicating a minimum value.
- If [tex]\( a < 0 \)[/tex], the parabola opens downwards, indicating a maximum value.
Since [tex]\( a = 3 > 0 \)[/tex], the parabola opens upwards, and thus the function has a minimum value.
3. Find the x-coordinate of the vertex (the point where the minimum or maximum occurs):
The formula for the x-coordinate of the vertex is given by:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
Plugging in the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[
x = -\frac{3}{2 \cdot 3} = -\frac{3}{6} = -\frac{1}{2}
\][/tex]
4. Find the y-coordinate of the vertex by substituting the x-coordinate back into the function [tex]\( f(x) \)[/tex]:
[tex]\[
y = f\left(-\frac{1}{2}\right)
\][/tex]
Substitute [tex]\( x = -\frac{1}{2} \)[/tex] into the equation [tex]\( f(x) = 3x^2 + 3x - 9 \)[/tex]:
[tex]\[
f\left(-\frac{1}{2}\right) = 3\left(-\frac{1}{2}\right)^2 + 3\left(-\frac{1}{2}\right) - 9
\][/tex]
Calculate each term separately:
[tex]\[
3\left(-\frac{1}{2}\right)^2 = 3 \cdot \frac{1}{4} = \frac{3}{4}
\][/tex]
[tex]\[
3\left(-\frac{1}{2}\right) = -\frac{3}{2}
\][/tex]
Combine these with the constant term (-9):
[tex]\[
y = \frac{3}{4} - \frac{3}{2} - 9
\][/tex]
Convert [tex]\(-\frac{3}{2}\)[/tex] to quarters to add easily:
[tex]\[
-\frac{3}{2} = -\frac{6}{4}
\][/tex]
Combine all terms:
[tex]\[
y = \frac{3}{4} - \frac{6}{4} - 9 = -\frac{3}{4} - 9 = -\frac{3}{4} - \frac{36}{4} = -\frac{39}{4}
\][/tex]
Therefore, the function has a minimum value. The coordinates of the minimum point are:
[tex]\[
\left( -\frac{1}{2}, -\frac{39}{4} \right)
\][/tex]
Thus, the correct answer is:
D. minimum; [tex]\(\left( -\frac{1}{2}, -\frac{39}{4} \right)\)[/tex]
