To address which function has the greatest value when [tex]\( x > 1 \)[/tex], let's examine and compare the values of the given functions at [tex]\( x = 1.1 \)[/tex]:
1. Function [tex]\( g(x) = 2 \cdot 5^x \)[/tex]
First, we evaluate [tex]\( g(x) \)[/tex] at [tex]\( x = 1.1 \)[/tex]:
[tex]\[
g(1.1) = 2 \cdot 5^{1.1}
\][/tex]
Plugging in the value, we find:
[tex]\[
g(1.1) \approx 11.746
\][/tex]
2. Function [tex]\( f(x) = 2x + 5 \)[/tex]
Next, we evaluate [tex]\( f(x) \)[/tex] at [tex]\( x = 1.1 \)[/tex]:
[tex]\[
f(1.1) = 2 \cdot 1.1 + 5
\][/tex]
Simplifying, we get:
[tex]\[
f(1.1) = 2.2 + 5 = 7.2
\][/tex]
3. Function [tex]\( h(x) = 2x^2 + 5 \)[/tex]
Then, we evaluate [tex]\( h(x) \)[/tex] at [tex]\( x = 1.1 \)[/tex]:
[tex]\[
h(1.1) = 2 \cdot (1.1)^2 + 5
\][/tex]
Calculating further,
[tex]\[
h(1.1) = 2 \cdot 1.21 + 5 = 2.42 + 5 = 7.42
\][/tex]
4. Function [tex]\( k(x) = 2x^3 + 5 \)[/tex]
Finally, we evaluate [tex]\( k(x) \)[/tex] at [tex]\( x = 1.1 \)[/tex]:
[tex]\[
k(1.1) = 2 \cdot (1.1)^3 + 5
\][/tex]
Calculating further,
[tex]\[
k(1.1) = 2 \cdot 1.331 + 5 \approx 2.662 + 5 = 7.662
\][/tex]
Now, let's compare all these values:
- [tex]\( g(1.1) \approx 11.746 \)[/tex]
- [tex]\( f(1.1) = 7.2 \)[/tex]
- [tex]\( h(1.1) = 7.42 \)[/tex]
- [tex]\( k(1.1) \approx 7.662 \)[/tex]
The highest value among these is [tex]\( g(1.1) \approx 11.746 \)[/tex].
Hence, the function [tex]\( g(x) = 2 \cdot 5^x \)[/tex] has the greatest value when [tex]\( x > 1 \)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{1}
\][/tex]
