Questions: Find the absolute maximum and minimum values of the function over the indicated interval, and indicate the x-values at which they occur. f(x)=4 x+9 ;[-6,5] The absolute maximum value is at x= . (Use a comma to separate answers as needed.)

Solution Steps

To find the absolute maximum and minimum values of a continuous function on a closed interval, evaluate the function at critical points and endpoints. Since the function \( f(x) = 4x + 9 \) is linear, it has no critical points. Therefore, evaluate the function at the endpoints of the interval \([-6, 5]\).

We evaluate the function \( f(x) = 4x + 9 \) at the endpoints of the interval \([-6, 5]\):

• At \( x = -6 \): \[ f(-6) = 4(-6) + 9 = -24 + 9 = -15 \]
• At \( x = 5 \): \[ f(5) = 4(5) + 9 = 20 + 9 = 29 \]

From the evaluations:

• The function value at \( x = -6 \) is \( -15 \).
• The function value at \( x = 5 \) is \( 29 \).

Thus, the absolute maximum value is \( 29 \) at \( x = 5 \), and the absolute minimum value is \( -15 \) at \( x = -6 \).

Final Answer