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 \( (x, y)= \) \( \square \) \( 1) \)

To find the critical numbers of \( f(x) = x^2 - 6x \), we first calculate its derivative: \[ f'(x) = 2x - 6. \] Setting the derivative equal to zero gives us the critical numbers: \[ 2x - 6 = 0 \implies x = 3. \] So, the critical number is \( x = 3 \). For intervals of increase and decrease, we can analyze the sign of the derivative. We observe: - \( f'(x) < 0 \) for \( x < 3 \) (decreasing). - \( f'(x) > 0 \) for \( x > 3 \) (increasing). Thus, in interval notation: - Increasing: \( (3, \infty) \) - Decreasing: \( (-\infty, 3) \) Now, applying the First Derivative Test to identify relative extrema, we know that at \( x = 3 \), the function changes from decreasing to increasing, indicating a relative minimum. To find the corresponding \( y \)-value: \[ f(3) = 3^2 - 6(3) = 9 - 18 = -9. \] Hence: - Relative maximum: \( DNE \) (there are no relative maxima). - Relative minimum: \( (x, y) = (3, -9) \). Now, let’s dive into a bit more context and problem-solving strategies! The vertex form of a quadratic function, such as this one, can make understanding its behavior much easier. By rewriting \( f(x) = x^2 - 6x \) in vertex form, completing the square reveals that the vertex is at \( (3, -9) \), confirming our earlier results on the minima and maxima! When analyzing critical numbers, always remember to verify the end behavior of the function. For this function, as \( x \) approaches \( \infty \) or \( -\infty \), \( f(x) \) approaches \( \infty \), indicating that there are no relative maxima but only a relative minimum, which is a classic feature of parabolas that open upwards!