Q. 80 - Calculus

Question

Sketch careful, labeled graphs of each function f in Exercises 63–82 by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f, f\text{'}, and f\text{'}\text{'},$ and examine any relevant limits so that you can describe all key points and behaviors of f. 

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

Step-by-Step Solution

Answer

The sign chart is

The sketch of the graph is

Step 1. Given Information.

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

Step 2. Finding the roots.

To find the roots we will put the given function equal to zero.

So,

$$\begin{gathered} f\left(x\right)=\cos \left(3\left(x-\frac{\mathrm{\pi }}{2}\right)\right) \\ 0=\cos \left(3\left(x-\frac{\mathrm{\pi }}{2}\right)\right) \\ x=0,\frac{2\mathrm{\pi }}{3} \end{gathered}$$

Therefore, the given function has roots at $x=0,\frac{2\mathrm{\pi }}{3}.$

Step 3. Testing the signs.

Now, let's test the sign for $f\text{'} and f\text{'}\text{'}.$

Let's differentiate the equation to find $f\text{'}.$

So,

 $$\begin{gathered} f\text{'}\left(x\right)=-3\sin \left(3\left(x-\frac{\mathrm{\pi }}{2}\right)\right) \\ 0=-3\sin \left(3\left(x-\frac{\mathrm{\pi }}{2}\right)\right) \\ x=\frac{\mathrm{\pi }}{2} \end{gathered}$$

Thus, $f\text{'}$ has a local maximum at $x=\frac{\mathrm{\pi }}{2}.$ It is positive on the interval $\left(-\infty ,\frac{\mathrm{\pi }}{2}\right)$. Hence the graph of f will be increasing during the positive interval.

Let's differentiate again.

So, 

$f\text{'}\text{'}\left(x\right)=-9\cos \left(3\left(x-\frac{\mathrm{\pi }}{2}\right)\right)$

Thus, $f\text{'}\text{'}$ has no inflection point. It is negative everywhere. Hence, the graph of f will be concave down everywhere.

Step 4. Sketch the sign chart.

The sign chart is 

Step 5. Examine the relevant limit.

Let's examine the limits.

$$\begin{gathered} \underset{x\to \infty }{\lim }f\left(x\right)= \mathrm{diverge} \\ \underset{\mathrm{x}\to -\infty }{\lim }\mathrm{f}\left(\mathrm{x}\right)=\mathrm{diverge} \end{gathered}$$

Step 6. Sketch the graph of function f.

The graph of the function is