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 $f,f\text{'},andf\text{'}\text{'},$ and examine any relevant limits so that you can describe all key points and behaviors of f.

$f\left(x\right)=\frac{1}{x^2+3}$

Short answer

The sign chart is

The sketch of the graph is

Step-by-step solution

Step 1. Given Information. 

The given function is$f\left(x\right)=\frac{1}{x^2+3}.$

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

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

So,

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

Let's examine the limits

$$\begin{gathered} \underset{x\to \infty }{\lim }f\left(x\right)=\frac{1}{x^2+3} \\ \underset{x\to \infty }{\lim }f\left(x\right)=0 \\ And \\ \underset{x\to -\infty }{\lim }f\left(x\right)=\frac{1}{x^2+3} \\ \underset{x\to -\infty }{\lim }f\left(x\right)=0 \end{gathered}$$

Step 3. Testing the signs.

To sketch the sign chart, let's differentiate the equation to find $f\text{'}.$

So,

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

Testing the signs on both sides,

$$\begin{gathered} f\text{'}\left(5\right)=\frac{5^2-2\left(5\right)+3}{{\left(5^2+3\right)}^2} \\ f\text{'}\left(5\right)=\frac{18}{784} \\ \mathrm{And} \\ f\text{'}(-2)=\frac{{(-2)}^2-2(-2)+3}{{\left({(-2)}^2+3\right)}^2} \\ f\text{'}(-2)=\frac{11}{49} \\ \mathrm{And} \\ f\text{'}\left(0\right)=\frac{0^2-2\left(0\right)+3}{{\left(0^2+3\right)}^2} \\ f\text{'}\left(0\right)=\frac{3}{9} \end{gathered}$$

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

Step 4. Testing the signs. 

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

Let's differentiate again.

So,

$f\text{'}\text{'}\left(x\right)=\frac{\left(2x-2\right){\left(x^2+3\right)}^2-\left(x^2-2x+3\right)\left(4x\left(x^2+3\right)\right)}{{\left(x^2+3\right)}^4}$

Thus, $f\text{'}\text{'}$is positive. Hence fwill be concave up everyplace.

Step 5. Sketch the sign chart. 

The sign chart is

Step 6. Sketch the graph of function f. 

The graph of the function is