Questions: Find the absolute maximum and minimum values of the following function on the given interval. Then graph the function. f(x) = (8/9)x - 3, -1 ≤ x ≤ 0 Find the absolute maximum value. Select the correct choice below and, if necessary, fill in the answer boxes to complete A. The absolute maximum value occurs at x= . (Simplify your answers. Use a comma to separate answers as needed.) B. There is no absolute maximum.

Solution Steps

The function given is \( f(x) = \frac{8}{9}x - 3 \) and the interval is \(-1 \leq x \leq 0\).

Evaluate \( f(x) \) at the endpoints of the interval to find the absolute maximum and minimum values.

• At \( x = -1 \): \[ f(-1) = \frac{8}{9}(-1) - 3 = -\frac{8}{9} - 3 = -\frac{8}{9} - \frac{27}{9} = -\frac{35}{9} \approx -3.8889 \]

• At \( x = 0 \): \[ f(0) = \frac{8}{9}(0) - 3 = -3 \]

Compare the values obtained at the endpoints:

• \( f(-1) \approx -3.8889 \)
• \( f(0) = -3 \)

The absolute maximum value is \(-3\) at \(x = 0\), and the absolute minimum value is approximately \(-3.8889\) at \(x = -1\).

Final Answer