Q. 65 - 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 $f$.

$f\left(x\right)={\left(1-x^4\right)}^7$.

Step-by-Step Solution

Answer

The graph for the function $f\left(x\right)={\left(1-x^4\right)}^7$ is,

Step 1 . Given information

$f\left(x\right)={\left(1-x^4\right)}^7$.

Step 2 . Let y = 1 - x 4 7 .

Now point the table for the function is given by,

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

Step 3 . The graph for the function is,

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

$$\begin{gathered} \frac{d}{dx}{\left(1-x^4\right)}^7=0 \\ 7{\left(1-x^4\right)}^6\left(-4x^3\right)=0 \\ {x}^3{\left(1-x^4\right)}^6=0 \\ x=0\text{ or }\left(1-x^2\right)\left(1+x^2\right)=0 \\ x=0\text{ or }1-x^2=0 \\ x=0\text{ or }{x}^2=1 \\ x=0,\pm 1 \end{gathered}$$

Therefore, $f$ has a critical point at $x=0,\pm 1$. But that is a local maximum at $x=0$.

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

Therefore, the function $f$ is increasing on $\left(-\infty ,0\right)$ and decreasing on $\left(0,\infty \right)$.

Again,

$$\begin{gathered} \underset{x\to -\infty }{\lim }f\left(x\right)=\underset{x\to -\infty }{\lim }{\left(1-x^4\right)}^7 \\ =-\infty \\ \underset{x\to \infty }{\lim }f\left(x\right)=\underset{x\to \infty }{\lim }{\left(1-x^4\right)}^7 \\ =\infty \end{gathered}$$

Therefore, the function is defined at everywhere, roots at $x=\pm 1$. Positive on $\left(-1,1\right)$ and negative elsewhere. And the limits are $\underset{x\to -\infty }{\lim }f\left(x\right)=-\infty ;\underset{x\to \infty }{\lim }f\left(x\right)=-\infty$.