Answer the questions below about the quadratic function.

[tex]\[ f(x) = -3x^2 + 30x - 78 \][/tex]

1. Does the function have a minimum or maximum value?
- Minimum
- Maximum

2. Where does the minimum or maximum value occur?
[tex]\[ x = \square \][/tex]

3. What is the function's minimum or maximum value?
[tex]\[ \square \][/tex]



Answer :

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].