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 $f,f\text{'},andf\text{'}\text{'},$ and examine any relevant limits so that you can describe all key points and behaviors of f.

$f\left(x\right)=\sqrt{x}\left(4-x\right)$

Short answer

The sign chart is

The sketch of the graph is

Step-by-step solution

Step 1. Given Information.  

The given function is$f\left(x\right)=\sqrt{x}\left(4-x\right).$

Step 2. Finding the roots and examining the relevant limit.  

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

So,

$$\begin{gathered} f\left(x\right)=\sqrt{x}\left(4-x\right) \\ 0=\sqrt{x}\left(4-x\right) \\ x=0\mathrm{and}4-x=0 \\ 4=x \end{gathered}$$

The given function has roots at $0and4.$It is positive on the interval $\left(0,4\right)$and negative elsewhere.

Let's examine the limits.

$\underset{x\to \infty }{\lim }f\left(x\right)=-\infty$

Step 3. Testing the signs.

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

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

So,

$$\begin{gathered} f\text{'}\left(x\right)=-\frac{3x-4}{2\sqrt{x}} \\ 0=-\frac{3x-4}{2\sqrt{x}} \\ 0=-3x+4 \\ -4=-3x \\ x=\frac{4}{3} \end{gathered}$$

Thus, $f\text{'}$has a local maximum at $x=\frac{4}{3}.$It is positive on the interval $\left(0,\frac{4}{3}\right)$ and negative elsewhere. Hence the graph of fwill be increasing on the positive intervals and decreasing on the negative intervals.

Let's differentiate again.

So,

$$\begin{gathered} f\text{'}\text{'}\left(x\right)=-\frac{3x+4}{4x^{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}} \\ 0=-\frac{3x+4}{4x^{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}} \\ 0=-3x-4 \\ 4=-3x \\ -\frac{4}{3}=x \end{gathered}$$

Thus, $f\text{'}\text{'}$ is negative everywhere. Hence, the graph of fwill be concave down always.

Step 4. Sketch the sign chart. 

The sign chart is

Step 5. Sketch the graph of function f.  

The graph of the function is