Q. 81S - 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 $f$.

$f\left(x\right)=e^{x^3-3x^2+2x}$

Step-by-Step Solution

Answer

The function is increasing on $(-\infty ,0.422)\cup (1.577,\infty )$ and decreasing on $(0.422,1.577)$

The local maxima $x=0.422$ and local minima $x=1.577$

The function is defined everywhere, except at $x=0$. It has no roots.

The limits of the function $\underset{x\to \infty }{\lim }f\left(x\right)=\infty$ and $\underset{x\to -\infty }{\lim }f\left(x\right)=0$.

Step 1: Given information

Given,$f\left(x\right)=e^{x^3-3x^2+2x}$

Step 2: Simplification

Consider $f\left(x\right)=e^{x^3-3x^2+2x}$

Make a table of values

$x$	$-2$	$-1$	$0$	$2$
$f\left(x\right)$	$0$	$0$	$1$	$1$

Draw the graph by hand as follows:

Find the Critical points by equating the first derivative to zero.

$\begin{array}{rcl}f\left(x\right)& =& e^{x^3-3x^2+2x}\\ f^{\text{'}}\left(x\right)& =& e^{x^3-3x^2+2x}\left(3x^2-6x+2\right)=0\\ x& =& 1\pm \frac{1}{\sqrt{3}}\end{array}$

The sign diagram of derivative:

Find the limits of the function

$$\begin{gathered} \underset{x\to \infty }{\lim }f\left(x\right)=\underset{x\to \infty }{\lim }{e}^{x^3-3x^2+2x}=\infty \\ \underset{x\to -\infty }{\lim }f\left(x\right)=\underset{x\to -\infty }{\lim }{e}^{x^3-3x^2+2x}=0 \end{gathered}$$

Hence, the function is increasing on $(-\infty ,0.422)\cup (1.577,\infty )$ and decreasing on $(0.422,1.577)$. The local maxima $x=0.422$ and local minima $x=1.577$. The function is defined everywhere, except at $x=0$. It has no roots.

The limits of the function $\underset{x\to \infty }{\lim }f\left(x\right)=\infty$ and $\underset{x\to -\infty }{\lim }f\left(x\right)=0$.