To determine the behavior of the function [tex]\( f(x) = \sqrt[3]{x+8} \)[/tex], we need to analyze its derivative. The derivative will help us understand whether the function is increasing or decreasing.
Step-by-Step Solution:
1. Define the Function:
The function given is [tex]\( f(x) = \sqrt[3]{x+8} \)[/tex].
2. Compute the Derivative:
We need to find the first derivative [tex]\( f'(x) \)[/tex] of the function.
Recall the formula for the derivative of a power function [tex]\( (x^n)' = nx^{n-1} \)[/tex]:
[tex]\[
f(x) = (x + 8)^{1/3}
\][/tex]
Using the chain rule:
[tex]\[
f'(x) = \frac{1}{3} (x + 8)^{-2/3}
\][/tex]
Simplifying the expression:
[tex]\[
f'(x) = \frac{1}{3} \cdot \frac{1}{(x + 8)^{2/3}}
\][/tex]
[tex]\[
f'(x) = \frac{1}{3 (x + 8)^{2/3}}
\][/tex]
3. Analyze the Derivative:
The expression for [tex]\( f'(x) \)[/tex] is:
[tex]\[
f'(x) = \frac{1}{3 (x + 8)^{2/3}}
\][/tex]
Notice that:
- The denominator [tex]\( (x + 8)^{2/3} \)[/tex] is always positive for all real [tex]\( x \)[/tex].
- The numerator is a positive constant [tex]\( \frac{1}{3} \)[/tex].
Therefore, the derivative [tex]\( f'(x) \)[/tex] is always positive for all [tex]\( x \in \mathbb{R} \)[/tex].
4. Conclusion:
Since the derivative [tex]\( f'(x) \)[/tex] is always positive, the function [tex]\( f(x) \)[/tex] is always increasing for all [tex]\( x \)[/tex].
Given this analysis, the correct statement is:
The function is always increasing.
