Let's solve the problem step-by-step:
1. Define the functions:
- The function [tex]\( h(x) \)[/tex] is given by:
[tex]\[
h(x) = 2(x - 7)^2
\][/tex]
- The function [tex]\( k(x) \)[/tex] is given by:
[tex]\[
k(x) = x^4 + 2x^2 + 8x - 4
\][/tex]
2. Evaluate [tex]\( h(x) \)[/tex] and [tex]\( k(x) \)[/tex] for given [tex]\( x \)[/tex] values:
- The provided [tex]\( x \)[/tex] values are: [tex]\( -2, -1, 0, 1, 2, 3 \)[/tex]
3. Evaluate [tex]\( h(x) \)[/tex] for each [tex]\( x \)[/tex]:
- [tex]\( h(-2) = 2(-2 - 7)^2 = 2(-9)^2 = 2 \times 81 = 162 \)[/tex]
- [tex]\( h(-1) = 2(-1 - 7)^2 = 2(-8)^2 = 2 \times 64 = 128 \)[/tex]
- [tex]\( h(0) = 2(0 - 7)^2 = 2(-7)^2 = 2 \times 49 = 98 \)[/tex]
- [tex]\( h(1) = 2(1 - 7)^2 = 2(-6)^2 = 2 \times 36 = 72 \)[/tex]
- [tex]\( h(2) = 2(2 - 7)^2 = 2(-5)^2 = 2 \times 25 = 50 \)[/tex]
- [tex]\( h(3) = 2(3 - 7)^2 = 2(-4)^2 = 2 \times 16 = 32 \)[/tex]
4. Evaluate [tex]\( k(x) \)[/tex] for each [tex]\( x \)[/tex]:
- [tex]\( k(-2) = (-2)^4 + 2(-2)^2 + 8(-2) - 4 = 16 + 8 - 16 - 4 = 4 \)[/tex]
- [tex]\( k(-1) = (-1)^4 + 2(-1)^2 + 8(-1) - 4 = 1 + 2 - 8 - 4 = -9 \)[/tex]
- [tex]\( k(0) = 0^4 + 2(0)^2 + 8(0) - 4 = -4 \)[/tex]
- [tex]\( k(1) = 1^4 + 2(1)^2 + 8(1) - 4 = 1 + 2 + 8 - 4 = 7 \)[/tex]
- [tex]\( k(2) = 2^4 + 2(2)^2 + 8(2) - 4 = 16 + 8 + 16 - 4 = 36 \)[/tex]
- [tex]\( k(3) = 3^4 + 2(3)^2 + 8(3) - 4 = 81 + 18 + 24 - 4 = 119 \)[/tex]
5. Identify the minimum values of each function:
- The minimum value of [tex]\( h(x) \)[/tex] from the computed values is [tex]\( 32 \)[/tex].
- The minimum value of [tex]\( k(x) \)[/tex] from the computed values is [tex]\( -9 \)[/tex].
6. Determine which function has the least and greatest minimum value:
- The function with the least minimum value is [tex]\( k(x) \)[/tex] because [tex]\( -9 \)[/tex] is less than [tex]\( 32 \)[/tex].
- The function with the greatest minimum value is [tex]\( h(x) \)[/tex] because [tex]\( 32 \)[/tex] is greater than [tex]\( -9 \)[/tex].
Based on these steps, we can complete the statements:
- The function that has the least minimum value is function [tex]\( k(x) \)[/tex].
- The function that has the greatest minimum value is function [tex]\( h(x) \)[/tex].
