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.)

$f\left(x\right)=\cos \left(\pi x\right)$

Short answer

The function has local minimum at all odd integers and local maximum at all the even integers.

Step-by-step solution

Step 1. Given Information.

The given function is$f\left(x\right)=\cos \left(\pi x\right)$

Step 2. Second-Derivative Test

On differentiating the given function, we have,

$$\begin{gathered} f^{\text{'}}\left(x\right)=\frac{d}{dx}\cos \left(\pi x\right) \\ =-\sin \left(\pi x\right)\frac{d}{dx}\left(\pi x\right) \\ =-\pi \sin \left(\pi x\right) \\ Now, \\ {f}^{\text{'}\text{'}}\left(x\right)=-\pi \frac{d}{dx}\sin \left(\pi x\right) \\ =-{\pi }^2\cos \left(\pi x\right) \\ Also, \\ f<0atalloddintegers.[LocalMaximum] \\ f>0atallevenintegers[LocalMinimum] \end{gathered}$$

Step 3. Verification.

The graph of the function is ,