Answer :
To solve the problem of finding the vertex and determining whether the function has a maximum or minimum value, we start by analyzing the quadratic function given:
[tex]\[ f(x) = -4x^2 + 24x - 112 \][/tex]
### Step 1: Find the Vertex
The vertex of a quadratic function in the form [tex]\( f(x) = ax^2 + bx + c \)[/tex] can be found using the formula for the x-coordinate of the vertex:
[tex]\[ x = -\frac{b}{2a} \][/tex]
For our function, [tex]\( a = -4 \)[/tex], [tex]\( b = 24 \)[/tex], and [tex]\( c = -112 \)[/tex].
#### Calculate the x-coordinate of the vertex:
[tex]\[ x = -\frac{24}{2(-4)} = -\frac{24}{-8} = 3 \][/tex]
#### Calculate the y-coordinate of the vertex:
We substitute [tex]\( x = 3 \)[/tex] back into the function to find the y-coordinate:
[tex]\[ f(3) = -4(3)^2 + 24(3) - 112 \][/tex]
[tex]\[ f(3) = -4(9) + 72 - 112 \][/tex]
[tex]\[ f(3) = -36 + 72 - 112 \][/tex]
[tex]\[ f(3) = 36 - 112 \][/tex]
[tex]\[ f(3) = -76 \][/tex]
So, the vertex of the function is [tex]\( (3, -76) \)[/tex].
### Step 2: Determine Maximum or Minimum Value
The function [tex]\( f(x) = ax^2 + bx + c \)[/tex] takes a maximum value if [tex]\( a < 0 \)[/tex] and a minimum value if [tex]\( a > 0 \)[/tex].
Since [tex]\( a = -4 \)[/tex] in our case, which is less than 0, the function has a maximum value.
### Conclusion
The vertex of the function is [tex]\( (3, -76) \)[/tex], and since the parabola opens downwards, the maximum value of the function is [tex]\(-76\)[/tex].
Thus, the answer is:
- Vertex: [tex]\( (3, -76) \)[/tex]
- Maximum value: [tex]\(-76\)[/tex]
[tex]\[ f(x) = -4x^2 + 24x - 112 \][/tex]
### Step 1: Find the Vertex
The vertex of a quadratic function in the form [tex]\( f(x) = ax^2 + bx + c \)[/tex] can be found using the formula for the x-coordinate of the vertex:
[tex]\[ x = -\frac{b}{2a} \][/tex]
For our function, [tex]\( a = -4 \)[/tex], [tex]\( b = 24 \)[/tex], and [tex]\( c = -112 \)[/tex].
#### Calculate the x-coordinate of the vertex:
[tex]\[ x = -\frac{24}{2(-4)} = -\frac{24}{-8} = 3 \][/tex]
#### Calculate the y-coordinate of the vertex:
We substitute [tex]\( x = 3 \)[/tex] back into the function to find the y-coordinate:
[tex]\[ f(3) = -4(3)^2 + 24(3) - 112 \][/tex]
[tex]\[ f(3) = -4(9) + 72 - 112 \][/tex]
[tex]\[ f(3) = -36 + 72 - 112 \][/tex]
[tex]\[ f(3) = 36 - 112 \][/tex]
[tex]\[ f(3) = -76 \][/tex]
So, the vertex of the function is [tex]\( (3, -76) \)[/tex].
### Step 2: Determine Maximum or Minimum Value
The function [tex]\( f(x) = ax^2 + bx + c \)[/tex] takes a maximum value if [tex]\( a < 0 \)[/tex] and a minimum value if [tex]\( a > 0 \)[/tex].
Since [tex]\( a = -4 \)[/tex] in our case, which is less than 0, the function has a maximum value.
### Conclusion
The vertex of the function is [tex]\( (3, -76) \)[/tex], and since the parabola opens downwards, the maximum value of the function is [tex]\(-76\)[/tex].
Thus, the answer is:
- Vertex: [tex]\( (3, -76) \)[/tex]
- Maximum value: [tex]\(-76\)[/tex]
