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 7 at $x = 0$
(use a comma to separate answers as needed.)
(b) the absolute maximum value is $\square$ at $x=\square$
(use a comma to separate answers as needed.)

Explanation:

Step1: Analyze the function's derivative

The function \(f(x) = 6x+7\) has a derivative \(f^\prime(x)=6\). Since \(f^\prime(x)>0\) for all \(x\), the function is increasing on the entire real line.

Step2: Evaluate the function at the endpoints of the interval \([-2,6]\)

• For \(x = - 2\):

\(f(-2)=6\times(-2)+7=-12 + 7=-5\)

• For \(x = 6\):

\(f(6)=6\times6+7=36 + 7=43\)

Answer:

The absolute maximum value is \(43\) at \(x = 6\). The absolute minimum value is \(-5\) at \(x=-2\).