Answered

IDNLearn.com makes it easy to find accurate answers to your specific questions. Find reliable solutions to your questions quickly and easily with help from our experienced experts.

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]

Sagot :

GinnyAnsweridn
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].
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Your questions are important to us at IDNLearn.com. Thanks for stopping by, and come back for more reliable solutions.