Answer :
To solve this problem, we need to determine the minimum value of the function [tex]\( f(x) = x^2 + 6x + 6 \)[/tex] and the maximum value of the function [tex]\( g(x) \)[/tex], and then find the difference between these two values.
### Step-by-Step Solution:
1. Find the minimum value of [tex]\( f(x) \)[/tex]:
- The function [tex]\( f(x) = x^2 + 6x + 6 \)[/tex] is a quadratic function. The minimum value of a quadratic function [tex]\( ax^2 + bx + c \)[/tex] can be found at its vertex.
- The vertex [tex]\( x \)[/tex] value is given by the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
- In the quadratic function [tex]\( f(x) = x^2 + 6x + 6 \)[/tex], we have [tex]\( a = 1 \)[/tex] and [tex]\( b = 6 \)[/tex].
[tex]\[ x = -\frac{6}{2 \cdot 1} = -3 \][/tex]
- Substitute [tex]\( x = -3 \)[/tex] back into [tex]\( f(x) \)[/tex] to find the minimum value:
[tex]\[ f(-3) = (-3)^2 + 6(-3) + 6 = 9 - 18 + 6 = -3 \][/tex]
So, the minimum value of [tex]\( f(x) \)[/tex] is [tex]\(-3\)[/tex].
2. Find the maximum value of [tex]\( g(x) \)[/tex]:
- Suppose from the graph of [tex]\( g(x) \)[/tex], the maximum value given is 27.
3. Calculate the difference between the maximum value of [tex]\( g(x) \)[/tex] and the minimum value of [tex]\( f(x) \)[/tex]:
- The minimum value of [tex]\( f(x) \)[/tex] is [tex]\(-3\)[/tex].
- The maximum value of [tex]\( g(x) \)[/tex] is [tex]\( 27 \)[/tex].
- The difference is:
[tex]\[ \text{Difference} = \text{Maximum of } g(x) - \text{Minimum of } f(x) = 27 - (-3) = 27 + 3 = 30 \][/tex]
So, the difference between the maximum value of [tex]\( g(x) \)[/tex] and the minimum value of [tex]\( f(x) \)[/tex] is [tex]\( 30 \)[/tex].
### Final Answer:
The difference between the maximum value of [tex]\( g(x) \)[/tex] and the minimum value of [tex]\( f(x) \)[/tex] is [tex]\( 30 \)[/tex].
### Step-by-Step Solution:
1. Find the minimum value of [tex]\( f(x) \)[/tex]:
- The function [tex]\( f(x) = x^2 + 6x + 6 \)[/tex] is a quadratic function. The minimum value of a quadratic function [tex]\( ax^2 + bx + c \)[/tex] can be found at its vertex.
- The vertex [tex]\( x \)[/tex] value is given by the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
- In the quadratic function [tex]\( f(x) = x^2 + 6x + 6 \)[/tex], we have [tex]\( a = 1 \)[/tex] and [tex]\( b = 6 \)[/tex].
[tex]\[ x = -\frac{6}{2 \cdot 1} = -3 \][/tex]
- Substitute [tex]\( x = -3 \)[/tex] back into [tex]\( f(x) \)[/tex] to find the minimum value:
[tex]\[ f(-3) = (-3)^2 + 6(-3) + 6 = 9 - 18 + 6 = -3 \][/tex]
So, the minimum value of [tex]\( f(x) \)[/tex] is [tex]\(-3\)[/tex].
2. Find the maximum value of [tex]\( g(x) \)[/tex]:
- Suppose from the graph of [tex]\( g(x) \)[/tex], the maximum value given is 27.
3. Calculate the difference between the maximum value of [tex]\( g(x) \)[/tex] and the minimum value of [tex]\( f(x) \)[/tex]:
- The minimum value of [tex]\( f(x) \)[/tex] is [tex]\(-3\)[/tex].
- The maximum value of [tex]\( g(x) \)[/tex] is [tex]\( 27 \)[/tex].
- The difference is:
[tex]\[ \text{Difference} = \text{Maximum of } g(x) - \text{Minimum of } f(x) = 27 - (-3) = 27 + 3 = 30 \][/tex]
So, the difference between the maximum value of [tex]\( g(x) \)[/tex] and the minimum value of [tex]\( f(x) \)[/tex] is [tex]\( 30 \)[/tex].
### Final Answer:
The difference between the maximum value of [tex]\( g(x) \)[/tex] and the minimum value of [tex]\( f(x) \)[/tex] is [tex]\( 30 \)[/tex].
