To find [tex]\( f(0) \)[/tex] for the function [tex]\( f(x) = 2x^2 + 5 \sqrt{x + 2} \)[/tex], follow these steps:
1. Substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ f(x) = 2x^2 + 5 \sqrt{x + 2} \][/tex]
[tex]\[ f(0) = 2(0)^2 + 5 \sqrt{0 + 2} \][/tex]
2. Simplify the terms inside the function:
[tex]\[ f(0) = 2 \cdot 0 + 5 \sqrt{2} \][/tex]
[tex]\[ f(0) = 0 + 5 \sqrt{2} \][/tex]
3. Calculate [tex]\( 5 \sqrt{2} \)[/tex]:
[tex]\[ 5 \sqrt{2} \approx 5 \times 1.414213562 \][/tex]
[tex]\[ 5 \sqrt{2} \approx 7.07106781186 \][/tex]
4. Round the result to the nearest hundredths place:
[tex]\[ 7.07106781186 \approx 7.07 \][/tex]
Therefore, the value of [tex]\( f(0) \)[/tex] rounded to the nearest hundredths place is:
[tex]\[ f(0) = \boxed{7.07} \][/tex]
