To determine which of the following functions has the largest value when [tex]\( x = 5 \)[/tex]:
[tex]\[
\begin{align*}
p(x) &= 2x^2 + 4x + 11 \\
h(x) &= 4^x \\
s(x) &= 10x \\
\end{align*}
\][/tex]
Step-by-step evaluation of each function at [tex]\( x = 5 \)[/tex]:
1. Calculate [tex]\( p(5) \)[/tex]:
[tex]\[
p(5) = 2(5)^2 + 4(5) + 11
\][/tex]
First, calculate [tex]\( 2(5)^2 \)[/tex]:
[tex]\[
2(5)^2 = 2(25) = 50
\][/tex]
Next, calculate [tex]\( 4(5) \)[/tex]:
[tex]\[
4(5) = 20
\][/tex]
Add these results together along with the constant 11:
[tex]\[
p(5) = 50 + 20 + 11 = 81
\][/tex]
2. Calculate [tex]\( h(5) \)[/tex]:
[tex]\[
h(5) = 4^5
\][/tex]
Calculate [tex]\( 4^5 \)[/tex] through exponentiation:
[tex]\[
4^5 = 1024
\][/tex]
3. Calculate [tex]\( s(5) \)[/tex]:
[tex]\[
s(5) = 10(5)
\][/tex]
Calculate [tex]\( 10(5) \)[/tex]:
[tex]\[
10(5) = 50
\][/tex]
Summary of the values calculated at [tex]\( x = 5 \)[/tex]:
[tex]\[
\begin{align*}
p(5) &= 81 \\
h(5) &= 1024 \\
s(5) &= 50 \\
\end{align*}
\][/tex]
To find the largest value, compare these results:
[tex]\[
p(5) = 81, \quad h(5) = 1024, \quad s(5) = 50
\][/tex]
Among the values [tex]\( 81 \)[/tex], [tex]\( 1024 \)[/tex], and [tex]\( 50 \)[/tex], the largest value is [tex]\( 1024 \)[/tex].
Thus, the function with the largest value at [tex]\( x = 5 \)[/tex] is [tex]\( h(x) = 4^x \)[/tex].
