Questions: Find the absolute maximum and absolute minimum (if any) of the given function on the specified interval. f(x) = x^3 - 6x^2 + 5; -2 ≤ x ≤ 5

Solution Steps

To find the absolute maximum and minimum of the function \( f(x) = x^3 - 6x^2 + 5 \) on the interval \([-2, 5]\), we need to:

1. Evaluate the function at the endpoints of the interval.
2. Find the critical points by setting the derivative \( f'(x) \) to zero and solving for \( x \).
3. Evaluate the function at the critical points within the interval.
4. Compare the values obtained in steps 1 and 3 to determine the absolute maximum and minimum.

We are given the function \( f(x) = x^3 - 6x^2 + 5 \) and the interval \( [-2, 5] \).

The derivative of the function is calculated as follows: \[ f'(x) = 3x^2 - 12x \]

Setting the derivative equal to zero to find critical points: \[ 3x^2 - 12x = 0 \] Factoring gives: \[ 3x(x - 4) = 0 \] Thus, the critical points are: \[ x = 0 \quad \text{and} \quad x = 4 \]

We evaluate \( f(x) \) at the endpoints \( x = -2 \) and \( x = 5 \), as well as at the critical points \( x = 0 \) and \( x = 4 \):

• \( f(-2) = (-2)^3 - 6(-2)^2 + 5 = -8 - 24 + 5 = -27 \)
• \( f(5) = 5^3 - 6(5)^2 + 5 = 125 - 150 + 5 = -20 \)
• \( f(0) = 0^3 - 6(0)^2 + 5 = 5 \)
• \( f(4) = 4^3 - 6(4)^2 + 5 = 64 - 96 + 5 = -27 \)

The function values at the evaluated points are:

• \( f(-2) = -27 \)
• \( f(5) = -20 \)
• \( f(0) = 5 \)
• \( f(4) = -27 \)

The absolute minimum and maximum values are: \[ \text{Absolute Minimum} = -27, \quad \text{Absolute Maximum} = 5 \]

Final Answer