Q. 14 - Calculus

Question

Find the locations and values of any global extrema of each function f in Exercises 11–20 on each of the four given intervals. Do all work by hand by considering local extrema and endpoint behavior. Afterwards, check your answers with graphs.

$f\left(x\right)=\frac{3x-2}{x-1}$ on the intervals

$$\begin{gathered} \left(a\right) [0,2] \\ \left(b\right) [-2,0] \\ \left(c\right) [-1,1] \\ \left(d\right) (-1,1) \end{gathered}$$

Step-by-Step Solution

Answer

(a) The function $f\left(x\right)=\frac{3x-2}{x-1}$ have no global extrema since the function has no solutions.

Part (a) Step 1. Given Information.

The function: 

$$\begin{gathered} f\left(x\right)=\frac{3x-2}{x-1} \\ [0,2] \end{gathered}$$

Part (a) Step 2. Find the critical points.

The critical points are given by,
$$\begin{gathered} f\left(x\right)=\frac{3x-2}{x-1} \\ f\text{'}\left(x\right)=-\frac{1}{{(x-1)}^2} \end{gathered}$$

So,

 $$\begin{gathered} f\text{'}\left(x\right)=0 \\ -\frac{1}{{(x-1)}^2}=0 \\ x\ne 0 \end{gathered}$$

Part (a) Step 3. Refer the above observation.

From the above observation, the function has no solutions.

So the function does not have any local maximum and local minimum.

So the function don't have global extrema.