Chapter 3: Q. 74 (page 261)

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 $f^{'}$ and examine any relevant limits so that you can describe all key points and behaviors of $f$ .
$f (x) = \frac{x^{3}}{x^{2} - 3 x + 2}$ .

Short Answer

Expert verified

The graph for the function $f (x) = \frac{x^{3}}{x^{2} - 3 x + 2}$ is,

Step by step solution

Step 1. Given information

$f (x) = \frac{x^{3}}{x^{2} - 3 x + 2}$ .

Step 2. Let fx=x3x2-3x+2.

Now point table for the function is given by,

$x$	$y$	$(x, y)$
$- 3$	$- \frac{27}{20}$	$(- 3, - \frac{27}{20})$
$- 2$	$- \frac{2}{3}$	$(- 2, - \frac{2}{3})$
$- 1$	$- \frac{1}{6}$	$(- 1, - \frac{1}{6})$
$0$	$(0, 0)$	$(0, 0)$
$\frac{1}{2}$	role="math" localid="1648546939513" $\frac{1}{6}$	$(\frac{1}{2}, \frac{1}{6})$

Step 3. The graph for the function is,

Step 4. Now for critical point f'x=0.

$\frac{d}{d x} (\frac{x^{3}}{x^{2} - 3 x + 2}) = 0 \frac{(x^{2} - 3 x + 2) 3 x^{2} - (x^{3}) 2 (2 x - 3)}{{(x^{2} - 3 x + 2)}^{2}} = 0 \frac{(3 x^{4} - 3 x^{3} + 6 x^{2}) - 4 x^{4} + 6 x^{3}}{{(x^{2} - 3 x + 2)}^{2}} = 0 \frac{- x^{4} + 3 x^{3} + 6 x^{2}}{{(x^{2} - 3 x + 2)}^{2}} = 0 - x^{2} (x^{2} + 3 x + 6) = 0 x = 0; x^{2} + 3 x + 6 = 0 x = 0; x = \frac{- 3 \pm \sqrt{3^{2} - 4 \cdot 1 \cdot 6}}{2 \cdot 1} x = 0; x = \frac{- 3 \pm \sqrt{9 - 24}}{2} x = 0; x = \frac{- 3 \pm \sqrt{9 - 24}}{2} \notin ℝ x = 0$

Therefore, $f$ has a critical point at $x = 0$ .It has no local extrema.

Step 5. The sign chart of f is shown below:

For roots of the function,

$\frac{x^{3}}{x^{2} - 3 x + 2} = 0 x^{3} = 0 x = 0$

Therefore, the function $f$ is defined everywhere except at $x = 1, x = 2$ , where there is a vertical asymptote. The function is positive on $(0, 1)$ and negative elsewhere. The function doesn't have a local extrema. The function is always increasing everywhere except where the function is undefined. No horizontal asymptotes.