Question

Curve sketching: For each function $f$that follows, construct

sign charts for $f,f^{\text{'}},\text{and}{f}^{\text{'}\text{'}},$if possible. Examine function

values or limits at any interesting values and at $\pm \infty$. Then

interpret this information to sketch a labeled graph of $f$.

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

Short answer

The plot is

Step-by-step solution

Step 1. Given information

An expression is given as$f\left(x\right)={\tan }^{-1}x$

Step 2. Concept used

Properties of derivatives:

- If$f^{\text{'}}$is positive, $f$is increasing.

- If $f^{\text{'}}$is negative, $f$is decreasing.

- If $f^{\text{'}}$is zero, $f$is constant.

- If $f^{\text{'}\text{'}}$is positive, is concave up and if $f^{\text{'}\text{'}}$is negative, then $f$is concave down.

Step 3. Finding values 

For extreme values we will first put function equal to zero then derivative equal to zero and then second derivative equal to zero.

$$\begin{gathered} f\left(x\right)={\tan }^{-1}x=0 \\ \Rightarrow x=0 \end{gathered}$$

First derivative,

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

It has no roots.

Second derivative,

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

Step 4. Plotting graph

We know critical values so we can plot the graph as,