Question

Use a sign chart for$f\text{'}\text{'}$ to determine the intervals on which each function f in Exercises 41–52 is concave up or concave down, and identify the locations of any inflection points. Then verify your algebraic answers with graphs from a calculator or graphing utility

$f\left(x\right)=3\cos \left(\frac{\pi }{2}x\right)+5$

Short answer

Inflection points are$x=1+4n,3+4n,$concave up on$(1+4n,3+4n)$and concave down on$(4n,1+4n),(3+4n,4+4n).$

Step-by-step solution

Sep 1. Given information.

The given function is$f\left(x\right)=3\cos \left(\frac{\pi }{2}x\right)+5.$

Step 2. Second derivative.

On differentiating, we get,

$$\begin{gathered} f^{\text{'}}\left(x\right)=\frac{d}{dx}\left(3\cos \left(\frac{\pi }{2}x\right)+5\right) \\ =-\frac{3\pi }{2}\sin \left(\frac{\pi }{2}x\right) \\ {f}^{\text{'}\text{'}}\left(x\right)=\frac{d}{dx}\left(-\frac{3\pi }{2}\sin \left(\frac{\pi }{2}x\right)\right) \\ =-\frac{3{\pi }^2}{4}\cos \left(\frac{\pi }{2}x\right) \end{gathered}$$

Step 3. Sign chart.

Now,

$f\text{'}\text{'}\left(x\right)=0atx=1+4n,3+4n.$

Inflection points: $x=1+4n,3+4n.$

Concave up: $(1+4n,3+4n)$

Concave down:$(4n,1+4n),(3+4n,4+4n)$

Step 4. Verification.

The graph of the function is :