Q. 64 - 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 $f$.

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

Step-by-Step Solution

Answer

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

Step 1 . Given information

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

Step 2 . Let y = 3 x 5 - 10 x 4 .

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, $\underset{x\to -\infty }{\lim }f\left(x\right)=-\infty ; \underset{x\to \infty }{\lim }f\left(x\right)=\infty$.