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 $f$.

$f\left(x\right)=3x^5-10x^4$.

Short answer

The graph for the function$f\left(x\right)=3x^5-10x^4$ is,

Step-by-step solution

Step 1. Given information

$f\left(x\right)=3x^5-10x^4$.

Step 2. Let y=3x5-10x4.

Now point the table for the function is given by,

$x$	$y$	$\left(x,y\right)$
$0$	$0$	$\left(0,0\right)$
$1$	$-7$	$\left(1,-7\right)$
$-1$	$-13$	$\left(-1,-13\right)$
$2$	$-64$	$\left(2,-64\right)$
$3$	$-81$	$\left(3,-81\right)$

Step 3. The graph for the function is,

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

$$\begin{gathered} \frac{d}{dx}\left(3x^5-10x^4\right)=0 \\ 15x^4-40x^3=0 \\ 3x^4-8x^3=0 \\ {x}^3(3x-8)=0 \\ x=0,\frac{8}{3} \end{gathered}$$

Therefore,$f$has a critical point at$x=0,\frac{8}{3}$. But that is a local maximum at$x=0$and local minimum$x=\frac{8}{3}$.

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

Therefore the function $f$is increasing on $\left(-\infty ,0\right)$and $\left(\frac{8}{3},\infty \right)$which may also written as, $\left(-\infty ,0\right)\cup \left(\frac{8}{3},\infty \right)$that is $f$decreasing on $\left(0,\frac{8}{3}\right)$.

Again,

$$\begin{gathered} \underset{x\to -\infty }{\lim }f\left(x\right)=\underset{x\to -\infty }{\lim }\left(3x^5-10x^4\right) \\ =-\infty \\ \underset{x\to \infty }{\lim }f\left(x\right)=\underset{x\to \infty }{\lim }\left(3x^5-10x^4\right) \\ =\infty \end{gathered}$$

Therefore, the function is defined at everywhere, roots are$x=0,\frac{10}{3}$. And the limits are,$\underset{x\to -\infty }{\lim }f\left(x\right)=-\infty ;\underset{x\to \infty }{\lim }f\left(x\right)=\infty$.