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 $\frac{29}{7}$ occurs at $x = 1$.
(simplify your answers. use a comma to separate answers as needed.)
b. there is no absolute maximum.
find the absolute minimum value. select the correct choice below and, if necessary, fill in the answer boxes to complete your choice
a. the absolute minimum value occurs at $x =$.
(simplify your answers. use a comma to separate answers as needed.)
b. there is no absolute minimum.

Explanation:

Step1: Analyze the function's derivative

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

Step2: Evaluate the function at the endpoints

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

Answer:

The absolute minimum value \(\frac{27}{7}\) occurs at \(x=-1\).