Q. 37 - Calculus

Question

Use the second-derivative test to determine the local extrema of each function $f$ in Exercises 29–40. If the second-derivative test fails, you may use the first-derivative test. Then verify your algebraic answers with graphs from a calculator or graphing utility. (Note: These are the same functions that you examined with the first-derivative test in Exercises 39–50 of Section 3.2.) 

$f\left(x\right)=\arctan x$

Step-by-Step Solution

Answer

There is no local extrema.

Step 1. Given Information.

The given function is $f\left(x\right)=\arctan x$.

Step 2. Critical point.

Differentiating the function, we get,

$$\begin{gathered} f^{\text{'}}\left(x\right)=\frac{d}{dx}\arctan \left(\pi x\right) \\ =\frac{1}{1+(\pi x{)}^2}\frac{d}{dx}\left(\pi x\right) \\ =\frac{\pi }{1+(\pi x{)}^2} \end{gathered}$$

This derivative of the function does not have any point where it is zero. Thus it has no critical point. Thus the function has no local extrema.

Step 3. Verification.

The graph of the function is,