Answered

Consider continuous functions [tex]\( f, g, h, \)[/tex] and [tex]\( k \)[/tex]. Then complete the statements.

[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-2 & 76 \\
-1 & 11 \\
0 & -4 \\
1 & -5 \\
2 & -4 \\
3 & 11 \\
\hline
\end{array}
\][/tex]

Function [tex]\( h \)[/tex] is two times the square of the difference of [tex]\( x \)[/tex] and 1:
[tex]\[ h(x) = 2(x - 1)^2 \][/tex]

Function [tex]\( k \)[/tex] is given by:
[tex]\[ k(x) = x^4 + 2x^2 + 8x - 4 \][/tex]

Select the correct answer from each drop-down:

The function that has the least minimum value is function [tex]\( \square \)[/tex]

The function that has the greatest minimum value is function [tex]\( \square \)[/tex]



Answer :

GinnyAnswer
To solve this problem and complete the given statements, follow these steps:

### Step 1: Determine the minimum value of the function [tex]\( h(x) \)[/tex]
#### Definition:
[tex]\[ h(x) = 2(x - 1)^2 \][/tex]
To find the minimum value, analyze the function:

1. The function [tex]\( (x - 1)^2 \)[/tex] is always non-negative and has its minimum value when [tex]\( x = 1 \)[/tex].
2. At [tex]\( x = 1 \)[/tex]:
[tex]\[ h(1) = 2(1 - 1)^2 = 2 \cdot 0 = 0 \][/tex]

Thus, the minimum value of [tex]\( h(x) \)[/tex] is [tex]\( 0 \)[/tex].

### Step 2: Determine the minimum value of the function [tex]\( k(x) \)[/tex]
#### Definition:
[tex]\[ k(x) = x^4 + 2x^2 + 8x - 4 \][/tex]
To find the minimum value of [tex]\( k(x) \)[/tex]:

1. Evaluate [tex]\( k(x) \)[/tex] over a suitable range, or realize that its minimum value from the given computations (for a range like [tex]\( -100 \)[/tex] to [tex]\( 100 \)[/tex]) is:
[tex]\[ k(x) = -9 \][/tex]

Thus, the minimum value of [tex]\( k(x) \)[/tex] is [tex]\( -9 \)[/tex].

### Step 3: Determine the minimum value of the function [tex]\( f(x) \)[/tex]
#### Table of values:
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -2 & 76 \\ -1 & 11 \\ 0 & -4 \\ 1 & -5 \\ 2 & -4 \\ 3 & 11 \\ \hline \end{array} \][/tex]

From the table, the minimum value of [tex]\( f(x) \)[/tex] is clearly:
[tex]\[ -5 \][/tex]

### Step 4: Compare the minimum values of [tex]\( h(x) \)[/tex], [tex]\( k(x) \)[/tex], and [tex]\( f(x) \)[/tex]
The minimum values found are:
- [tex]\( h(x) \)[/tex]: [tex]\( 0 \)[/tex]
- [tex]\( k(x) \)[/tex]: [tex]\( -9 \)[/tex]
- [tex]\( f(x) \)[/tex]: [tex]\( -5 \)[/tex]

### Least Minimum Value
The function with the least (smallest) minimum value is [tex]\( k(x) \)[/tex] because [tex]\( -9 \)[/tex] (minimum of [tex]\( k(x) \)[/tex]) is less than [tex]\( -5 \)[/tex] (minimum of [tex]\( f(x) \)[/tex]) and [tex]\( 0 \)[/tex] (minimum of [tex]\( h(x) \)[/tex]).

### Greatest Minimum Value
The function with the greatest minimum value is [tex]\( h(x) \)[/tex] because [tex]\( 0 \)[/tex] (minimum of [tex]\( h(x) \)[/tex]) is greater than [tex]\( -5 \)[/tex] (minimum of [tex]\( f(x) \)[/tex]) and [tex]\( -9 \)[/tex] (minimum of [tex]\( k(x) \)[/tex]).

### Final Statements
- The function that has the least minimum value is function [tex]\( k \)[/tex].
- The function that has the greatest minimum value is function [tex]\( h \)[/tex].