Consider the following function. Use a graphing utility to confirm your answers for parts (a) through (c). (If an answer does not exist, enter DNE.) \[ f(x)=|x+2|-3 \] (a) Find the critical numbers of \( f \). (Enter your answers as a comma-separated llist.) \[ x=\square \] (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 relative extrema. relative maximum \( \quad(x, y)=( \) \( \square \) relative minimum \[ (x, y)=(\square) \]

To solve the given function \( f(x) = |x + 2| - 3 \): First, we need to find the critical numbers. The critical points occur where the derivative is zero or undefined. 1. The function can be split into two cases based on the absolute value: - For \( x + 2 \geq 0 \) (i.e., \( x \geq -2 \)): \( f(x) = x + 2 - 3 = x - 1 \) - For \( x + 2 < 0 \) (i.e., \( x < -2 \)): \( f(x) = - (x + 2) - 3 = -x - 5 \) 2. Next, we find the derivative in both cases: - If \( x \geq -2 \): \( f'(x) = 1 \) - If \( x < -2 \): \( f'(x) = -1 \) The critical number occurs where the function changes from one case to the other, which is at \( x = -2 \). So the critical numbers of \( f \) are: \[ x = -2 \] Now, let's determine where the function is increasing or decreasing: - The function is increasing where the derivative is positive: \( f'(x) = 1 \) implies it's increasing for \( x \geq -2 \). - The function is decreasing where the derivative is negative: \( f'(x) = -1 \) implies it's decreasing for \( x < -2 \). Thus, the interval notation is: - Increasing: \( (-2, \infty) \) - Decreasing: \( (-\infty, -2) \) Finally, we apply the First Derivative Test: - For \( x < -2 \), \( f \) is decreasing, and at \( x = -2 \), \( f \) changes from decreasing to increasing. - This indicates a relative minimum. Calculating \( f(-2) \): \[ f(-2) = |0| - 3 = -3 \] Therefore, the relative minimum is: \[ (x, y) = (-2, -3) \] In summary: (a) \( x = -2 \) (b) increasing \( (-2, \infty) \), decreasing \( (-\infty, -2) \) (c) relative maximum \( \quad(x, y)=(\text{none}) \) relative minimum \( (x, y)=(-2, -3) \)