Q. 35 - 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.) 

$f\left(x\right)=\cos \left(\pi \right(x+1\left)\right)$

Step-by-Step Solution

Answer

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

Step 1. Given Information.

The given function is $f\left(x\right)=\cos \left(\pi \right(x+1\left)\right).$

Step 2. Second-Derivative Test.

On differentiating the given function, we get,

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

Again differentiating, we get,

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

Now,

$$\begin{gathered} f<0 for all odd integers.\{Local maximium\} \\ f>0 for all even integers.\{Local minimum\} \end{gathered}$$

Step 3. Verification.

The graph of the function is,