Consider the following function. Use a graphing utillty to confirm your answers for parts (a) through (c). (lf an answer does not exist, enter DNE.) \[ f(x)=x^{1 / 5}+2 \] (a) Find the critical numbers of \( f \). (Enter your answers as a comma-separated Ilst.) \[ x=\square \] (b) Find the open intervals on which the function is increasing or decreasing. (Enter your answers using interval notation.) increasing \( \square \) decreasing \( \square \) (c) Apply the First Derivative Test to Identify all ralative extrema. relative maximum \( \quad(x, y)=(\square \) \( \square \) relatlve minimum \( \quad(x, y)=( \) \( \square \) )

To find the critical numbers of the function \( f(x)=x^{1/5}+2 \), we first need to compute the derivative \( f'(x) \). The derivative can be calculated as follows: \[ f'(x) = \frac{1}{5}x^{-4/5} \] Setting the derivative equal to zero gives us: \[ \frac{1}{5}x^{-4/5} = 0 \] Since \( x^{-4/5} \) can never be zero, there are no critical numbers. The function is defined for all real numbers; thus, we can conclude: x = DNE To determine whether \( f \) is increasing or decreasing, we look at the sign of \( f'(x) \): - For \( x > 0 \): \( f'(x) > 0 \) (increasing) - For \( x < 0 \): \( f'(x) < 0 \) (decreasing) - At \( x = 0 \): The derivative does not exist. Thus, we can state the open intervals: increasing \( (0, \infty) \) decreasing \( (-\infty, 0) \) Now for the First Derivative Test, since there is no critical number, the function does not have any relative extrema. However, we observe that as the function transitions from decreasing to increasing at \( x=0 \), we can say that there is a vertical tangent but not a traditional maximum or minimum: relative maximum \( \quad(x, y)=(DNE) \quad DNE \) relative minimum \( \quad(x, y)=(0, 2) \)