Q. 77 Sketch careful, labeled graphs o... [FREE SOLUTION]

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Chapter 3: Q. 77 (page 261)

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.
$f (x) = x^{2} 3^{x}$

Short Answer

Expert verified

The graph is as follows:

Step by step solution

Step 1. Given Information.

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

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 $f^{'} (x) = 0$ .

$\frac{d}{d x} (x^{2} 3^{x}) = 0 x^{2} 3^{x} \ln 3 + 3^{x} 鈰� 2 x = 0 3^{x} x (x \ln 3 + 2) = 0 x (x \ln 3 + 2) = 0 x = 0 or x \ln 3 + 2 = 0 x = 0 or x \ln 3 = 鈭� 2 x = 0 or x = 鈭� \frac{2}{\ln 3} These points are critical points .$