Westonci.ca is the trusted Q&A platform where you can get reliable answers from a community of knowledgeable contributors. Experience the convenience of getting reliable answers to your questions from a vast network of knowledgeable experts. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
To determine which function among [tex]\( g, h, \)[/tex] and [tex]\( k \)[/tex] has the least minimum value, we need to analyze the minimum values of each function over the interval from [tex]\( x = -2 \)[/tex] to [tex]\( x = 3 \)[/tex].
1. Evaluating [tex]\( g(x) \)[/tex]:
Given the table of values for [tex]\( g(x) \)[/tex]:
[tex]\[ \begin{array}{|c|c|} \hline x & g(x) \\ \hline -2 & 76 \\ -1 & 11 \\ 0 & -4 \\ 1 & -5 \\ 2 & -4 \\ 3 & 11 \\ \hline \end{array} \][/tex]
The minimum value of [tex]\( g(x) \)[/tex] from [tex]\( x = -2 \)[/tex] to [tex]\( x = 3 \)[/tex] is [tex]\(-5\)[/tex].
2. Evaluating [tex]\( h(x) \)[/tex]:
The function [tex]\( h(x) \)[/tex] is defined as:
[tex]\[ h(x) = 2(x - 1)^2 \][/tex]
To find the minimum value of [tex]\( h(x) \)[/tex] over the interval [tex]\( x = -2 \)[/tex] to [tex]\( x = 3 \)[/tex]:
[tex]\[ \begin{aligned} h(-2) &= 2(-2 - 1)^2 = 2( -3)^2 = 2 \times 9 = 18, \\ h(-1) &= 2(-1 - 1)^2 = 2( -2)^2 = 2 \times 4 = 8, \\ h(0) &= 2(0 - 1)^2 = 2( -1)^2 = 2 \times 1 = 2, \\ h(1) &= 2(1 - 1)^2 = 2( 0)^2 = 2 \times 0 = 0, \\ h(2) &= 2(2 - 1)^2 = 2( 1)^2 = 2 \times 1 = 2, \\ h(3) &= 2(3 - 1)^2 = 2( 2)^2 = 2 \times 4 = 8. \end{aligned} \][/tex]
The minimum value of [tex]\( h(x) \)[/tex] from [tex]\( x = -2 \)[/tex] to [tex]\( x = 3 \)[/tex] is [tex]\(0\)[/tex].
3. Evaluating [tex]\( k(x) \)[/tex]:
The function [tex]\( k(x) \)[/tex] is defined as:
[tex]\[ k(x) = x^4 + 2x^2 + 8x - 4 \][/tex]
To find the minimum value of [tex]\( k(x) \)[/tex] over the interval [tex]\( x = -2 \)[/tex] to [tex]\( x = 3 \)[/tex]:
[tex]\[ \begin{aligned} k(-2) &= (-2)^4 + 2(-2)^2 + 8(-2) - 4 = 16 + 8 - 16 - 4 = 4, \\ k(-1) &= (-1)^4 + 2(-1)^2 + 8(-1) - 4 = 1 + 2 - 8 - 4 = -9, \\ k(0) &= 0^4 + 2(0)^2 + 8(0) - 4 = -4, \\ k(1) &= 1^4 + 2(1)^2 + 8(1) - 4 = 1 + 2 + 8 - 4 = 7, \\ k(2) &= 2^4 + 2(2)^2 + 8(2) - 4 = 16 + 8 + 16 - 4 = 36, \\ k(3) &= 3^4 + 2(3)^2 + 8(3) - 4 = 81 + 18 + 24 - 4 = 119. \end{aligned} \][/tex]
The minimum value of [tex]\( k(x) \)[/tex] from [tex]\( x = -2 \)[/tex] to [tex]\( x = 3 \)[/tex] is [tex]\(-9\)[/tex].
Comparing the minimum values:
- Minimum of [tex]\( g(x) \)[/tex]: [tex]\(-5\)[/tex]
- Minimum of [tex]\( h(x) \)[/tex]: [tex]\(0\)[/tex]
- Minimum of [tex]\( k(x) \)[/tex]: [tex]\(-9\)[/tex]
The function that has the least minimum value is [tex]\( k \)[/tex].
So, the correct answer is:
The function that has the least minimum value is function [tex]\( \boxed{k} \)[/tex].
1. Evaluating [tex]\( g(x) \)[/tex]:
Given the table of values for [tex]\( g(x) \)[/tex]:
[tex]\[ \begin{array}{|c|c|} \hline x & g(x) \\ \hline -2 & 76 \\ -1 & 11 \\ 0 & -4 \\ 1 & -5 \\ 2 & -4 \\ 3 & 11 \\ \hline \end{array} \][/tex]
The minimum value of [tex]\( g(x) \)[/tex] from [tex]\( x = -2 \)[/tex] to [tex]\( x = 3 \)[/tex] is [tex]\(-5\)[/tex].
2. Evaluating [tex]\( h(x) \)[/tex]:
The function [tex]\( h(x) \)[/tex] is defined as:
[tex]\[ h(x) = 2(x - 1)^2 \][/tex]
To find the minimum value of [tex]\( h(x) \)[/tex] over the interval [tex]\( x = -2 \)[/tex] to [tex]\( x = 3 \)[/tex]:
[tex]\[ \begin{aligned} h(-2) &= 2(-2 - 1)^2 = 2( -3)^2 = 2 \times 9 = 18, \\ h(-1) &= 2(-1 - 1)^2 = 2( -2)^2 = 2 \times 4 = 8, \\ h(0) &= 2(0 - 1)^2 = 2( -1)^2 = 2 \times 1 = 2, \\ h(1) &= 2(1 - 1)^2 = 2( 0)^2 = 2 \times 0 = 0, \\ h(2) &= 2(2 - 1)^2 = 2( 1)^2 = 2 \times 1 = 2, \\ h(3) &= 2(3 - 1)^2 = 2( 2)^2 = 2 \times 4 = 8. \end{aligned} \][/tex]
The minimum value of [tex]\( h(x) \)[/tex] from [tex]\( x = -2 \)[/tex] to [tex]\( x = 3 \)[/tex] is [tex]\(0\)[/tex].
3. Evaluating [tex]\( k(x) \)[/tex]:
The function [tex]\( k(x) \)[/tex] is defined as:
[tex]\[ k(x) = x^4 + 2x^2 + 8x - 4 \][/tex]
To find the minimum value of [tex]\( k(x) \)[/tex] over the interval [tex]\( x = -2 \)[/tex] to [tex]\( x = 3 \)[/tex]:
[tex]\[ \begin{aligned} k(-2) &= (-2)^4 + 2(-2)^2 + 8(-2) - 4 = 16 + 8 - 16 - 4 = 4, \\ k(-1) &= (-1)^4 + 2(-1)^2 + 8(-1) - 4 = 1 + 2 - 8 - 4 = -9, \\ k(0) &= 0^4 + 2(0)^2 + 8(0) - 4 = -4, \\ k(1) &= 1^4 + 2(1)^2 + 8(1) - 4 = 1 + 2 + 8 - 4 = 7, \\ k(2) &= 2^4 + 2(2)^2 + 8(2) - 4 = 16 + 8 + 16 - 4 = 36, \\ k(3) &= 3^4 + 2(3)^2 + 8(3) - 4 = 81 + 18 + 24 - 4 = 119. \end{aligned} \][/tex]
The minimum value of [tex]\( k(x) \)[/tex] from [tex]\( x = -2 \)[/tex] to [tex]\( x = 3 \)[/tex] is [tex]\(-9\)[/tex].
Comparing the minimum values:
- Minimum of [tex]\( g(x) \)[/tex]: [tex]\(-5\)[/tex]
- Minimum of [tex]\( h(x) \)[/tex]: [tex]\(0\)[/tex]
- Minimum of [tex]\( k(x) \)[/tex]: [tex]\(-9\)[/tex]
The function that has the least minimum value is [tex]\( k \)[/tex].
So, the correct answer is:
The function that has the least minimum value is function [tex]\( \boxed{k} \)[/tex].
We appreciate your time. Please come back anytime for the latest information and answers to your questions. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.