Questions: Consider g(x)=2(x-6)(x-12)^2 Determine the x-intercept(s) of g. Report solutions in (x, y) form. Determine the y-intercept of g. Report solutions in (x, y) form. Determine the interval(s) on which g is decreasing. Determine the interval(s) on which g is increasing. Determine the value and location of any local minimum of f. Enter the solution in (x, g(x)) form. If multiple solutions exist, use a comma-separated list to enter the solutions. Determine the value and location of any local maximum of f. Enter the solution in (x, g(x)) form. If multiple solutions exist, use a comma-separated list to enter the solutions.

Solution Steps

To solve the given problems, we need to analyze the function \( g(x) = 2(x-6)(x-12)^2 \).

1. Finding the x-intercepts: The x-intercepts occur where \( g(x) = 0 \). This means solving the equation \( 2(x-6)(x-12)^2 = 0 \).

2. Finding the y-intercept: The y-intercept occurs where \( x = 0 \). Substitute \( x = 0 \) into the function to find \( g(0) \).

3. Determining intervals of increase and decrease: To find where the function is increasing or decreasing, we need to find the derivative \( g'(x) \) and determine where it is positive (increasing) or negative (decreasing).

To find the x-intercepts of the function \( g(x) = 2(x-6)(x-12)^2 \), we set \( g(x) = 0 \): \[ 2(x-6)(x-12)^2 = 0 \] This gives us the solutions \( x = 6 \) and \( x = 12 \). Therefore, the x-intercepts are: \[ (6, 0) \quad \text{and} \quad (12, 0) \]

The y-intercept occurs when \( x = 0 \). We calculate: \[ g(0) = 2(0-6)(0-12)^2 = 2(-6)(144) = -1728 \] Thus, the y-intercept is: \[ (0, -1728) \]

To find where \( g \) is increasing or decreasing, we analyze the derivative \( g'(x) \). The critical points are found to be \( x = 8 \) and \( x = 12 \). The intervals are determined as follows:

• \( g \) is increasing on \( (-\infty, 8) \) and \( (12, \infty) \).
• \( g \) is decreasing on \( (8, 12) \).

Final Answer