Q. 60 - 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 $f$.

$f\left(x\right)=x^3-9x+1$.

Step-by-Step Solution

Answer

The graph for the function $f\left(x\right)=x^3-9x+1$ is,

Step 1 . Given information

$f\left(x\right)=x^3-9x+1$.

Step 2 . Consider the equation,

$f\left(x\right)=x^3-9x+1$

Let $y=x^3-9x+1$

Now point the table for the function is given by,

$x$	$y$	$\left(x,y\right)$
$0$	$1$	$\left(0,1\right)$
$2$	$-9$	$\left(2,-9\right)$
$-1$	$9$	$\left(-1,9\right)$
$3$	$1$	$\left(3,1\right)$
$-2$	$11$	$\left(-2,11\right)$

Step 3 . The graph for the function is,

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

$$\begin{gathered} \frac{d}{dx}\left(x^3-9x+1\right)=0 \\ 3x^2-9=0 \\ {x}^2-3=0 \\ {x}^2=3 \\ x=\pm \sqrt{3} \end{gathered}$$

Therefore, $f$ has a local minimum at $x=-\sqrt{3}$ and has a local maximum at $x=\sqrt{3}$.

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

Therefore the function $f$ is increasing on $\left(-\infty ,-\sqrt{3}\right)$ and $\left(\sqrt{3},\infty \right)$ decreasing on $\left(-\sqrt{3},\sqrt{3}\right)$.

Again,

$$\begin{gathered} \underset{x\to -\infty }{\lim }f\left(x\right)=\underset{x\to -\infty }{\lim }\left(x^3-9x+1\right) \\ =-\infty \\ \underset{x\to \infty }{\lim }f\left(x\right)=\underset{x\to \infty }{\lim }\left(x^3-9x+1\right) \\ =\infty \end{gathered}$$

Therefore, the function is defined everywhere, roots are $x=\pm 3.05409,x=0.112638$. Positive on $\left(-\infty ,-\frac{1}{3}\right)\cup \left(1,\infty \right)$ and negative elsewhere.

Local minimum at $x=-\sqrt{3}$ and local maximum at $x=\sqrt{3}$.

The function $f$ is increasing on $\left(-\infty ,-\sqrt{3}\right)$ and $\left(\sqrt{3},\infty \right)$ and decreasing on $\left(-\sqrt{3},\sqrt{3}\right)$.

The limits are, $\underset{x\to -\infty }{\lim }f\left(x\right)=-\infty$ and $\underset{x\to \infty }{\lim }f\left(x\right)=\infty$.$\left(-\infty ,-\sqrt{3}\right)$