The given quadratic function is:
[tex]\[ f(x) = -3x^2 + 30x - 78 \][/tex]
Let's analyze this function step-by-step:
1. Determine if the function has a minimum or maximum value:
The quadratic function [tex]\( f(x) = ax^2 + bx + c \)[/tex] forms a parabola. The coefficient [tex]\( a \)[/tex] (the coefficient of [tex]\( x^2 \)[/tex]) determines the direction of the parabola:
- If [tex]\( a \)[/tex] is positive, the parabola opens upwards, and the function has a minimum value.
- If [tex]\( a \)[/tex] is negative, the parabola opens downwards, and the function has a maximum value.
In this function, [tex]\( a = -3 \)[/tex], which is negative. Therefore, the function has a maximum value.
2. Find the x-coordinate of the vertex (where the maximum value occurs):
The x-coordinate of the vertex of a parabola [tex]\( ax^2 + bx + c \)[/tex] is given by the formula:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
For the given function:
[tex]\[
a = -3, \quad b = 30
\][/tex]
Substitute these values into the formula:
[tex]\[
x = -\frac{30}{2(-3)} = -\frac{30}{-6} = 5
\][/tex]
Thus, the maximum value occurs at [tex]\( x = 5 \)[/tex].
3. Find the maximum value of the function:
To find the maximum value of the function, substitute [tex]\( x = 5 \)[/tex] back into the function [tex]\( f(x) \)[/tex]:
[tex]\[
f(5) = -3(5)^2 + 30(5) - 78
\][/tex]
Simplify:
[tex]\[
f(5) = -3(25) + 150 - 78
\][/tex]
[tex]\[
f(5) = -75 + 150 - 78
\][/tex]
[tex]\[
f(5) = 75 - 78
\][/tex]
[tex]\[
f(5) = -3
\][/tex]
Therefore, the maximum value of the function is [tex]\(-3\)[/tex].
Summary:
1. The function has a maximum value.
2. The maximum value occurs at [tex]\( x = 5 \)[/tex].
3. The maximum value of the function is [tex]\(-3\)[/tex].
