Q. 72 - Calculus

Question

Sketch careful, labeled graphs of each function f in Exercises 63–82 by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f, f\text{'}, and f\text{'}\text{'},$ and examine any relevant limits so that you can describe all key points and behaviors of f. 

$f\left(x\right)=\frac{1}{{\left(x-1\right)}^2\left(x-2\right)}$

Step-by-Step Solution

Answer

The sign chart is  

The sketch of the graph is   

Step 1. Given Information.

The given function is $f\left(x\right)=\frac{1}{{\left(x-1\right)}^2\left(x-2\right)}.$

Step 2. Finding the roots and examining the relevant limit.

To find the roots we will put the given function equal to zero.

So,

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

The given function is undefined at $x=1,2.$ It has no roots.

Let's examine the limits.

$$\begin{gathered} \underset{x\to \infty }{\lim }f\left(x\right)=0 \\ \underset{x\to -\infty }{\lim }f\left(x\right)=0 \end{gathered}$$

Step 3. Testing the signs.

Now, let's test the sign for $f\text{'} \mathrm{and} f\text{'}\text{'}.$

Let's differentiate the equation to find $f\text{'}.$

So, 

 $$\begin{gathered} f\text{'}\left(x\right)=-\frac{3x-5}{{\left(x-2\right)}^2{\left(x-1\right)}^3} \\ 0=-\frac{3x-5}{{\left(x-2\right)}^2{\left(x-1\right)}^3} \\ 0=-3x+5 \\ 3x=5 \\ x=\frac{5}{3} \end{gathered}$$

Thus, $f\text{'}$ is positive on the interval  $\left(-\infty ,\infty \right).$Hence the graph of f will be increasing on the positive interval.

Let's differentiate again.

So, 

$$\begin{gathered} f\text{'}\text{'}=\frac{2\left(6x^2-20x+17\right)}{{\left(x-2\right)}^3{\left(x-1\right)}^4} \\ 0=\frac{2\left(6x^2-20x+17\right)}{{\left(x-2\right)}^3{\left(x-1\right)}^4} \\ 0=2\left(6x^2-20x+17\right) \end{gathered}$$

Thus, $f\text{'}\text{'}$ is positive everywhere and it has no inflection point. Hence, the graph of f will be concave up everywhere.

Step 4. Sketch the sign chart.

The sign chart is 

Step 5. Sketch the graph of function f.

The graph of the function is