Question

find the absolute maximum and minimum values of the following function on the given interval. then graph the function.

$f(x)=\frac{3}{5}x + 4$, $-2\leq x\leq1$

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 $\frac{3}{5}$ occurs at $x = 1$.
(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 function is \( f(x)=\frac{3}{5}x + 4 \). The derivative \( f^\prime(x)=\frac{3}{5}\). Since \( f^\prime(x)=\frac{3}{5}>0\), the function is increasing on the interval \([-2,1]\).

Step2: Evaluate the function at the endpoints

• When \( x=-2\), \( f(-2)=\frac{3}{5}\times(-2)+4=-\frac{6}{5}+4=\frac{-6 + 20}{5}=\frac{14}{5}=2.8\)
• When \( x = 1\), \( f(1)=\frac{3}{5}\times1+4=\frac{3+20}{5}=\frac{23}{5}=4.6\)

Answer:

A. The absolute maximum value \(\frac{23}{5}\) occurs at \(x = 1\)