Q. 59 - 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 $f$.

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

Step-by-Step Solution

Answer

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

Step 1 . Given information

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

Step 2 . Consider the equation,

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

Let $y=x^3-x^2-x+1$

Now point the table for the function is given by,

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

Step 3 . The graph of the function is,

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

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

Therefore, $f$ has a local minimum at $x=1$.

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

Therefore, the function $f$ is increasing on $\left(-\infty ,-\frac{1}{3}\right)$ and $\left(1,\infty \right)$ and decreasing on $\left(-\frac{1}{3},1\right)$.

Again,

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

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

The function $f$ is increasing on $\left(-\infty ,-\frac{1}{3}\right)\cup \left(1,\infty \right)$ and decreasing on $\left(-\frac{1}{3},1\right)$.

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