Get the answers you need at Westonci.ca, where our expert community is always ready to help with accurate information. Get quick and reliable solutions to your questions from a community of seasoned experts on our user-friendly platform. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A 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].
Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.
