Q. 78 - Calculus

Question

Sketch careful, labeled graphs of each function $f$ in Exercises $57-82$ 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 $f$ and $f\text{'}$, and examine any relevant limits so that you can describe all key points and behaviors of $f$. 

Step-by-Step Solution

Answer

The function $f$ is defined everywhere, and the root of the $f$ is at $x=0$; The $f$ is positive everywhere except on $x=0$. The $f$ has a local minimum at $x=-3$; and local maximum at $x=0$. The function is increasing on $(-3,\infty )$ and negative elsewhere. $\underset{x\to -\infty }{\lim }f\left(x\right)=0$. Therefore, there is horizontal asymptote on the left at $y=0.$$\underset{x\to \infty }{\lim }f\left(x\right)=\infty$; Hence, there is no horizontal asymptote on the right.

Step 1. Given data

We have the given function, $f\left(x\right)=x^3{e}^x$

that is,$y=x^3{e}^x$

Step 2. Graph

The point table of the function is given by,

$x$	$y$	(x,y)
$-2$	$-\frac{8}{e^2}$	$(-2,-\frac{8}{e^2})$
$-1$	$-\frac{1}{e}$	$(-1,-\frac{1}{e})$
$0$	0	$(0,0)$
$1$	$e$	$(1,e)$
$2$	$8e^2$	$(2,8e^2)$

 The graph of the function is, 

Step 3. Critical points

To find the critical points, let $f\text{'}\left(x\right)=0$

$\begin{array}{rl}\frac{d}{dx}\left(x^3{e}^x\right)& =0\\ x^3{e}^x+e^x\cdot 3x^2& =0\\ e^x{x}^2(x+3)& =0\\ x^2(x+3)& =0\end{array}$

That is,

$x^2=0$  or  $$\begin{gathered} (x+3)=0 \\ x=3 \end{gathered}$$

Hence, $f$ has a critical points are at $x=0,-3$.

Here, the Local minima at $x=-3$, and the local maxima at $x=0$.

Step 4. Sign chart

The sign chart of the function is,

Step 5. Roots of the function

For the roots of the functions be,

$\begin{array}{rl}{x}^3{e}^x& =0\\ x^3& =0\\ x& =0\end{array}$

Hence the function has a root

Again,

$\begin{array}{rl}\underset{x\to \mathrm{\infty }}{lim}f\left(x\right)& =\underset{x\to \mathrm{\infty }}{lim}{x}^3{e}^x\\ & =\mathrm{\infty }\\ \underset{x\to -\mathrm{\infty }}{lim}f\left(x\right)& =\underset{x\to -\mathrm{\infty }}{lim}{x}^3{e}^x\\ & =0\end{array}$

Hence, the $f$ is defined everywhere, and the root of the function is at $x=0$; The $f$ is positive everywhere except on $x=0$. The function has a local minimum at $x=-3$; and local maximum at $x=0$. The function is increasing on $(-3,\infty )$ and negative elsewhere. $\underset{x\to -\infty }{\lim }f\left(x\right)=0$. therefore, there is horizontal asymptote on the left at $y=0$ . $\underset{x\to \infty }{\lim }f\left(x\right)=\infty$; so there is no horizontal asymptote on the right.