Answer :
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]
### 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]
