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].
