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.) \[ f(x)=x^{2}-6 x \] (a) Find the critical numbers of \( f \). (Enter your answers as a comma-separated list.) \[ x=\square \] (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 extrerna. relative maximum \( \quad(x, y)=( \) \( \square \) relative minimum \( (x, y)= \) \( \square \) \( 1) \)

**Step 1. Find the Critical Numbers** The function is \[ f(x)=x^{2}-6x. \] First, we compute the derivative: \[ f'(x)=\frac{d}{dx}(x^{2}-6x)=2x-6. \] Critical numbers occur when \( f'(x)=0 \) or \( f'(x) \) is undefined. Since \( f'(x) \) is defined for all \( x \), we set \[ 2x-6=0. \] Solving for \( x \): \[ 2x=6 \quad \Rightarrow \quad x=3. \] Thus, the critical number is: \[ x=3. \] --- **Step 2. Determine Intervals of Increase and Decrease** The derivative is given by \[ f'(x)=2x-6. \] - **For \( x<3 \):** Choose a test point, say \( x=0 \). Then, \[ f'(0)=2(0)-6=-6<0. \] Since the derivative is negative, the function is decreasing on this interval. - **For \( x>3 \):** Choose a test point, say \( x=4 \). Then, \[ f'(4)=2(4)-6=8-6=2>0. \] Since the derivative is positive, the function is increasing on this interval. Thus, the function is: - Increasing on \((3, \infty)\). - Decreasing on \((-\infty, 3)\). --- **Step 3. Apply the First Derivative Test for Relative Extrema** At the critical number \( x=3 \), the derivative changes sign: - From negative (when \( x<3 \)) to positive (when \( x>3 \)). This sign change indicates a relative minimum at \( x=3 \). To find the \( y \)-coordinate of the relative minimum, substitute \( x=3 \) into \( f(x) \): \[ f(3)=3^{2}-6\cdot3=9-18=-9. \] Thus, there is a relative minimum at: \[ (x, y)=(3, -9). \] Since there is no change from positive to negative in \( f'(x) \), there is no relative maximum. --- **Final Answers** (a) Critical number: \( x=3 \). (b) Intervals: Increasing: \((3, \infty)\) Decreasing: \((-\infty, 3)\) (c) Relative extrema: Relative maximum: DNE Relative minimum: \( (3, -9) \)