To determine the relative extrema of the function [tex]\( f(x) = x^{2/3} - 6 \)[/tex], we will follow these steps:
1. Calculate the first derivative of [tex]\( f(x) \)[/tex]:
The first derivative [tex]\( f'(x) \)[/tex] helps us find the critical points.
[tex]\[
f(x) = x^{2/3} - 6
\][/tex]
Differentiating with respect to [tex]\( x \)[/tex]:
[tex]\[
f'(x) = \frac{2}{3}x^{-1/3}
\][/tex]
2. Find the critical points by setting the first derivative equal to zero and solving for [tex]\( x \)[/tex]:
[tex]\[
f'(x) = \frac{2}{3}x^{-1/3} = 0
\][/tex]
Since [tex]\( \frac{2}{3}x^{-1/3} = 0 \)[/tex] does not produce feasible solutions ([tex]\(x\)[/tex] cannot be zero for the derivative to be zero), we conclude that there are no critical points.
3. Examine the second derivative:
The second derivative [tex]\( f''(x) \)[/tex] is used in the Second Derivative Test to determine the concavity of the function at critical points.
[tex]\[
f''(x) = -\frac{2}{9}x^{-4/3}
\][/tex]
4. Analyze the second derivative:
Since there are no critical points, we don't need to plug any values into the second derivative. Consequently, we can't apply the Second Derivative Test to find relative extrema because there are no critical points to evaluate.
To summarize, based on the given function and given analysis, there are no relative extrema for the function [tex]\( f(x) = x^{2/3} - 6 \)[/tex].
Thus, we conclude:
- The relative maximum does not exist ([tex]\( DNE \)[/tex]).
- The relative minimum also does not exist ([tex]\( DNE \)[/tex]).
