Q. 57 - 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 $f$.

$f\left(x\right) = (x - 2)(3x + 1)$.

Step-by-Step Solution

Answer

The graph for the function $f\left(x\right) = (x - 2)(3x + 1)$ is,

Step 1 . Given information

$f\left(x\right) = (x - 2)(3x + 1)$.

Step 2 . Consider the equation,

$f\left(x\right) = (x - 2)(3x + 1)$.

Let $y = (x - 2)(3x + 1)$.

Now, point table for the function is given by,

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

The graph of the function is,

Step 3 . Now for the critical point f ' x = 0 .

$$\begin{gathered} \frac{d}{dx}\left(\right(x-2\left)\right(3x+1\left)\right)=0 \\ \frac{d}{dx}\left(3x^2+x-6x-2\right)=0 \\ \frac{d}{dx}\left(3x^2-5x-2\right)=0 \\ 6x-5=0 \\ 6x=5 \\ x=\frac{5}{6} \end{gathered}$$

Now $x=\frac{5}{6}$ lies on $\left(-\frac{1}{3},2\right)$ that is $f$ has a local minimum at $x=\frac{5}{6}$.

Step 4 . The sign chart for f is shown below.

Therefore, the function $f$ is increasing on $\left(\frac{5}{6},\infty \right)$ and decreasing on $\left(-\infty ,\frac{5}{6}\right)$.

Again,

$$\begin{gathered} \underset{x\to -\infty }{\lim }f\left(x\right)=\underset{x\to -\infty }{\lim }(x-2)(3x+1) \\ =\infty \\ \underset{x\to \infty }{\lim }f\left(x\right)=\underset{x\to \infty }{\lim }(x-2)(3x+1) \\ =\infty \end{gathered}$$

Step 5 . Therefore, the function is defined everywhere.

The roots are, $x=-\frac{1}{3},x=2$.

Positive on $\left(-\infty ,-\frac{1}{3}\right)\cup \left(2,\infty \right)$ and negative elsewhere. Local minimum at $x=\frac{5}{6}$.

The function $f$ is increasing on $\left(\frac{5}{6},\infty \right)$ and decreasing on $\left(-\infty ,\frac{5}{6}\right)$. And limits are $\underset{x\to -\infty }{\lim }f\left(x\right)=\infty$ and $\underset{x\to \infty }{\lim }f\left(x\right)=\infty$.