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Sagot :
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].
### 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].
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