Q. 74 - 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 $f$.

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

Step-by-Step Solution

Answer

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

Step 1 . Given information

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

Step 2 . Let f x = x 3 x 2 - 3 x + 2 .

Now point table for the function is given by,

$x$	$y$	$\left(x,y\right)$
$-3$	$-\frac{27}{20}$	$\left(-3,-\frac{27}{20}\right)$
$-2$	$-\frac{2}{3}$	$\left(-2,-\frac{2}{3}\right)$
$-1$	$-\frac{1}{6}$	$\left(-1,-\frac{1}{6}\right)$
$0$	$(0,0)$	$(0,0)$
$\frac{1}{2}$	$\frac{1}{6}$	$\left(\frac{1}{2},\frac{1}{6}\right)$

Step 3 . The graph for the function is,

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

$$\begin{gathered} \frac{d}{dx}\left(\frac{x^3}{x^2-3x+2}\right)=0 \\ \frac{\left(x^2-3x+2\right)3x^2-\left(x^3\right)2(2x-3)}{{\left(x^2-3x+2\right)}^2}=0 \\ \frac{\left(3x^4-3x^3+6x^2\right)-4x^4+6x^3}{{\left(x^2-3x+2\right)}^2}=0 \\ \frac{-x^4+3x^3+6x^2}{{\left(x^2-3x+2\right)}^2}=0 \\ -x^2\left(x^2+3x+6\right)=0 \\ x=0;x^2+3x+6=0 \\ x=0;x=\frac{-3\pm \sqrt{3^2-4·1·6}}{2·1} \\ x=0;x=\frac{-3\pm \sqrt{9-24}}{2} \\ x=0;x=\frac{-3\pm \sqrt{9-24}}{2}\notin \mathrm{ℝ} \\ x=0 \end{gathered}$$

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,

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

Therefore, the function $f$ is defined everywhere except at $x=1,x=2$, where there is a vertical asymptote. The function is positive on $\left(0,1\right)$ 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.