Question

find the absolute maximum and minimum values of the function over the indicated interval, and indicate the x - values at which they occur.
$f(x)=6x + 7$
(a) $0,9$(b) $-2,6$
(a) the absolute maximum value is 61 at $x = 9$
(use a comma to separate answers as needed.)
the absolute minimum value is $\square$ at $x=\square$
(use a comma to separate answers as needed.)

Explanation:

Step1: Analyze the function's monotonicity

The function \(f(x) = 6x + 7\) is a linear function with a slope \(m = 6>0\). A linear function \(y=mx + b\) (\(m>0\)) is increasing on the entire real - line.

Step2: Evaluate the function at the endpoints of the interval \([0,9]\)

For the left - hand endpoint \(x = 0\):
\(f(0)=6\times0 + 7=7\)
For the right - hand endpoint \(x = 9\):
\(f(9)=6\times9+7=54 + 7=61\)

Since the function is increasing, the minimum value occurs at the left - hand endpoint of the interval and the maximum value occurs at the right - hand endpoint of the interval.

Answer:

The absolute minimum value is \(7\) at \(x = 0\)