Q. 71 - 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. 

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

Step-by-Step Solution

Answer

The sign chart is  

The sketch of the graph is   

Step 1. Given Information.

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

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 $0 and 4.$ 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 f will 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 f will 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