Q. 77 - Calculus

Question

Sketch careful, labeled graphs of each function f 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' , and examine any relevant limits so that you can describe all key points and behaviors of f.

$f\left(x\right)=x^2{3}^x$

Step-by-Step Solution

Answer

The graph is as follows:

Step 1. Given Information.

We are given a function $f\left(x\right)=x^2{3}^x$.

We have to sketch a graph of this function by making sign charts for the signs, roots, and undefined points of f and f' , and examine any relevant limits to describe all key points and behaviors of f. 

Step 2. Table of Points

The table of points for given function is:

x	y	( x, y)
- 2	$\frac{4}{9}$	$(-2,\frac{4}{9})$
- 1	$\frac{1}{3}$	$(-1,\frac{1}{3})$
0	0	( 0, 0)
1	3	( 1, 3)
2	36	( 2, 36)

Step 3. Graphing the function

The graph is as follows:

Step 4. Make sign charts for the signs, roots, and undefined points of f and f'

Find critical point as $f^{\text{'}}\left(x\right)=0$.

$$\begin{gathered} \frac{d}{dx}\left(x^2{3}^x\right)=0 \\ {x}^2{3}^x\ln 3+3^x\cdot 2x=0 \\ {3}^{x}x(x\ln 3+2)=0 \\ x(x\ln 3+2)=0 \\ x=0\text{ or }x\ln 3+2=0 \\ x=0\text{ or }x\ln 3=-2 \\ x=0\text{ or }x=-\frac{2}{\ln 3} \\ \mathrm{These} \mathrm{points} \mathrm{are} \mathrm{critical} \mathrm{points}. \end{gathered}$$

The sign chart of f is as:

Find roots:

$$\begin{gathered} x^2{3}^x=0 \\ {x}^2=0\left[\text{ dividing both sides by }{3}^x\right] \\ x=0 \end{gathered}$$

Step 5. Find behaviors of f at extreme x.

$$\begin{gathered} \underset{x\to \mathrm{\infty }}{lim} f\left(x\right)=\underset{x\to \mathrm{\infty }}{lim} x^2{3}^x \\ =\mathrm{\infty } \\ \underset{\mathrm{x}\to -\mathrm{\infty }}{lim} \mathrm{f}\left(\mathrm{x}\right)=\underset{\mathrm{x}\to -\mathrm{\infty }}{lim} {\mathrm{x}}^2{3}^{\mathrm{x}} \\ =0 \end{gathered}$$

There is no horizontal asymptote on the right.