Question

For each function f that follows, construct sign charts forf, f',and f '', if possible. Examine function values or limits at any interesting values and at ±∞. Then interpret this information to sketch a labeled graph of f.

$f\left(x\right)=x\sqrt{x+1}.$

Short answer

sign charts forf, f',and f ''are following.

Graph of function is following.

Step-by-step solution

Step 1. Given information. 

The given function is$f\left(x\right)=x\sqrt{x+1}.$

Step 2. Roots of the function. 

Equate the function to zero.

$$\begin{gathered} f\left(x\right)=0 \\ x\sqrt{x+1}=0 \\ x=0,-1 \end{gathered}$$

The roots of the function are$x=0,-1.$

Step 3. critical points of the function. 

Determine the first derivative of the function.

$$\begin{gathered} f\text{'}\left(x\right)=\frac{d}{dx}\left(x\sqrt{x+1}\right) \\ =\frac{3x+2}{2\sqrt{x+1}} \end{gathered}$$

critical points of the function.

role="math" localid="1648669988461" $$\begin{gathered} f\text{'}\left(x\right)=0 \\ \frac{3x+2}{2\sqrt{x+1}}=0 \\ x=\frac{-2}{3} \end{gathered}$$

So function has the critical point at $x=\frac{-2}{3}.$

Step 4. inflection points of the function. 

find the second derivative of the function.

$$\begin{gathered} f\text{'}\text{'}\left(x\right)=\frac{d}{dx}\left(\frac{3x+2}{2\sqrt{x+1}}\right) \\ =\frac{3x+4}{4{\left(x+1\right)}^{{\displaystyle \frac{3}{2}}}} \end{gathered}$$

inflection points of the function

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

So the function has an inflection point at $x=\frac{-4}{3}.$

So the sign chart is the following.

Step 5. Graph of function. 

The graph of the function is the following.