Questions: Consider the function f(x) = 2x^3 + 15x^2 - 144x + 9 with -8 ≤ x ≤ 4 This function has an absolute minimum at the point and an absolute maximum at the point Note: both parts of this answer should be entered as an ordered pair, including the parentheses, such as (5,11).

Solution Steps

To find the absolute minimum and maximum of the function \( f(x) = 2x^3 + 15x^2 - 144x + 9 \) on the interval \([-8, 4]\), we need to:

1. Find the critical points by taking the derivative of \( f(x) \) and setting it to zero.
2. Evaluate \( f(x) \) at the critical points and at the endpoints of the interval.
3. Compare these values to determine the absolute minimum and maximum.

The function is given by

\[ f(x) = 2x^3 + 15x^2 - 144x + 9. \]

To find the critical points, we first compute the derivative:

\[ f'(x) = 6x^2 + 30x - 144. \]

Next, we set the derivative equal to zero to find the critical points:

\[ 6x^2 + 30x - 144 = 0. \]

Factoring or using the quadratic formula, we find the critical points to be

\[ x = -8 \quad \text{and} \quad x = 3. \]

We evaluate \( f(x) \) at the critical points and the endpoints of the interval \([-8, 4]\):

• At \( x = -8 \):

\[ f(-8) = 2(-8)^3 + 15(-8)^2 - 144(-8) + 9 = 1097. \]

• At \( x = 3 \):

\[ f(3) = 2(3)^3 + 15(3)^2 - 144(3) + 9 = -234. \]

• At \( x = 4 \):

\[ f(4) = 2(4)^3 + 15(4)^2 - 144(4) + 9 = -199. \]

Now we compare the values obtained:

• \( f(-8) = 1097 \)
• \( f(3) = -234 \)
• \( f(4) = -199 \)

The absolute minimum occurs at \( (3, -234) \) and the absolute maximum occurs at \( (-8, 1097) \).

Final Answer