Question

find the absolute maximum and minimum values of the following function on the given interval. then graph the function.
$f(x)=\frac{1}{7}x + 4$, $-1\leq x\leq1$
find the absolute maximum value. select the correct choice below and, if necessary, fill in the answer boxes to complete your choice
a. the absolute maximum value $\square$ occurs at $x = \square$.
(simplify your answers. use a comma to separate answers as needed.)
b. there is no absolute maximum.

Explanation:

Step1: Find the derivative of the function

The derivative of \(f(x)=\frac{1}{7}x + 4\) is \(f^\prime(x)=\frac{1}{7}\). Since \(f^\prime(x)=\frac{1}{7}>0\), the function is increasing on the interval \([-1,1]\).

Step2: Evaluate the function at the endpoints

Evaluate \(f(x)\) at \(x=-1\) and \(x = 1\).
When \(x=-1\), \(f(-1)=\frac{1}{7}\times(-1)+4=4-\frac{1}{7}=\frac{28 - 1}{7}=\frac{27}{7}\).
When \(x = 1\), \(f(1)=\frac{1}{7}\times1+4=4+\frac{1}{7}=\frac{28+1}{7}=\frac{29}{7}\).

Answer:

A. The absolute maximum value \(\frac{29}{7}\) occurs at \(x = 1\).