To solve this problem, we'll proceed with the following steps:
### Part (a)
We need to fill in the table by computing the values of the functions [tex]\( f(x) = 50x^2 \)[/tex] and [tex]\( g(x) = 4^x \)[/tex] for each given [tex]\( x \)[/tex].
Given in the table:
[tex]\[
\begin{array}{|c|c|c|}
\hline
x & f(x) = 50 x^2 & g(x) = 4^x \\
\hline
3 & 450 & 64 \\
\hline
\end{array}
\][/tex]
Let's compute the values for [tex]\( x =4, 5, 6, 7 \)[/tex]:
1. For [tex]\( x = 4 \)[/tex]:
[tex]\[
f(4) = 50 \cdot 4^2 = 50 \cdot 16 = 800
\][/tex]
[tex]\[
g(4) = 4^4 = 256
\][/tex]
2. For [tex]\( x = 5 \)[/tex]:
[tex]\[
f(5) = 50 \cdot 5^2 = 50 \cdot 25 = 1250
\][/tex]
[tex]\[
g(5) = 4^5 = 1024
\][/tex]
3. For [tex]\( x = 6 \)[/tex]:
[tex]\[
f(6) = 50 \cdot 6^2 = 50 \cdot 36 = 1800
\][/tex]
[tex]\[
g(6) = 4^6 = 4096
\][/tex]
4. For [tex]\( x = 7 \)[/tex]:
[tex]\[
f(7) = 50 \cdot 7^2 = 50 \cdot 49 = 2450
\][/tex]
[tex]\[
g(7) = 4^7 = 16384
\][/tex]
Now we fill in the table:
[tex]\[
\begin{tabular}{|c|c|c|}
\hline
x & f(x)=50 x^2 & g(x)=4^x \\
\hline
3 & 450 & 64 \\
\hline
4 & 800 & 256 \\
\hline
5 & 1250 & 1024 \\
\hline
6 & 1800 & 4096 \\
\hline
7 & 2450 & 16384 \\
\hline
\end{tabular}
\][/tex]
### Part (b)
We need to look at the values for [tex]\( x \geq 4 \)[/tex] and determine for which [tex]\( x \)[/tex] the function [tex]\( f(x) \)[/tex] is greater than [tex]\( g(x) \)[/tex].
From the table, for [tex]\( x \geq 4 \)[/tex]:
- For [tex]\( x = 4 \)[/tex], [tex]\( f(4) = 800 \)[/tex] and [tex]\( g(4) = 256 \)[/tex]. Here, [tex]\( f(4) > g(4) \)[/tex].
- For [tex]\( x = 5 \)[/tex], [tex]\( f(5) = 1250 \)[/tex] and [tex]\( g(5) = 1024 \)[/tex]. Here, [tex]\( f(5) > g(5) \)[/tex].
- For [tex]\( x = 6 \)[/tex], [tex]\( f(6) = 1800 \)[/tex] and [tex]\( g(6) = 4096 \)[/tex]. Here, [tex]\( f(6) < g(6) \)[/tex].
- For [tex]\( x = 7 \)[/tex], [tex]\( f(7) = 2450 \)[/tex] and [tex]\( g(7) = 16384 \)[/tex]. Here, [tex]\( f(7) < g(7) \)[/tex].
Therefore, for [tex]\( x \geq 4 \)[/tex], we observe that [tex]\( f(x) \)[/tex] is greater than [tex]\( g(x) \)[/tex] for [tex]\( x = 4 \)[/tex] and [tex]\( x = 5 \)[/tex], but [tex]\( f(x) \)[/tex] is less than [tex]\( g(x) \)[/tex] for [tex]\( x = 6 \)[/tex] and [tex]\( x = 7 \)[/tex].
Thus, the results for [tex]\( x \geq 4 \)[/tex] suggest that:
[tex]\[ \text{For } x \geq 4, \text{ the table suggests that } f(x) \text{ is greater than } g(x) \text{ for the values } x < 6. \][/tex]
The final statement can be rephrased as:
For [tex]\( x \geq 4 \)[/tex], the table suggests that [tex]\( f(x) \)[/tex] is:
[tex]\[ \boxed{\text{greater than } g(x) \text{ for some values and less for others.}} \][/tex]
