To determine the rate of increase for the function [tex]\( f(x) = \frac{1}{3}(\sqrt[3]{24})^{2x} \)[/tex], let's follow these steps:
1. Identify the base of the exponential function:
The given function is [tex]\( f(x) = \frac{1}{3}(\sqrt[3]{24})^{2x} \)[/tex]. The key part to focus on is the exponential component [tex]\((\sqrt[3]{24})^{2x}\)[/tex].
2. Calculate the cube root of 24:
The cube root of 24 is a specific value, approximately [tex]\( \sqrt[3]{24} \approx 2.8844991406148166 \)[/tex].
3. Raise the cube root to the power of 2:
To determine the rate of increase, we need to find the value of [tex]\((\sqrt[3]{24})^2\)[/tex].
So, [tex]\( \left(2.8844991406148166\right)^2 \approx 8.320335292207616 \)[/tex].
4. Identify the correct rate of increase:
Based on the given options:
- [tex]\(\frac{1}{3}\)[/tex]
- [tex]\(2 \sqrt[3]{3}\)[/tex]
- 4
- [tex]\(4 \sqrt[3]{9}\)[/tex]
The value [tex]\(\approx 8.320335292207616\)[/tex] best matches one of the given options. The correct expression that represents this value can be calculated as follows:
- Note that [tex]\(4 \cdot (\sqrt[3]{9})\)[/tex] can be simplified. The cube root of 9 is approximately [tex]\(2.080083823051904\)[/tex].
- Multiplying 4 by this cube root: [tex]\(4 \cdot 2.080083823051904 \approx 8.320335292207616\)[/tex].
Therefore, the rate of increase for the function [tex]\( f(x) = \frac{1}{3}(\sqrt[3]{24})^{2x} \)[/tex] is:
[tex]\[ \boxed{4 \sqrt[3]{9}} \][/tex]
