Q. 75 - 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 $f$.

$f\left(x\right)=x\ln x$.

Step-by-Step Solution

Answer

The graph for the function $f\left(x\right)=x\ln x$ is,

Step 1 . Given information

$f\left(x\right)=x\ln x$.

Step 2 . Let src="data:image/svg+xml;base64,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" role="math" localid="1648548589163" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/e3a39aa2-cffb-4e88-9920-a390b0fd0716.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220329%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220329T101855Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=979c14ce910f9452023d7a198bae8b26913addddbfca498d82c264ef2af661e8" y = x ln   x .

Now point table for the function is given by,

$x$	$y$	$\left(x,y\right)$
$0.0001$	$0$	$\left(0.0001,0\right)$
$1$	$0$	$\left(1,0\right)$
$2$	$1.386$	$\left(2,1.386\right)$
$3$	$3.296$	$\left(3,3.296\right)$

Step 3 . The graph for the function is,

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

$$\begin{gathered} \frac{d}{dx}\left(x\ln x\right)=0 \\ x·\frac{1}{x}+\ln x·1=0 \\ 1+\ln x=0 \\ \ln x=-1 \\ x=e^{-1} \end{gathered}$$

Therefore, $f$ has a critical point at $x=\frac{1}{e}$. It has a local minima at $x=\frac{1}{e}$.

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

For roots of the function,

$$\begin{gathered} x\ln x=0 \\ \ln x=0 \\ x=e^0 \\ x=1 \end{gathered}$$

Step 6 . Therefore, the function f is defined on 0 , ∞ .

The function is positive on $\left(1,\infty \right)$ and negative elsewhere it is defined. The function have a local minimum at $x=e^{-1}$. The function is increasing on $\left(\frac{1}{e},\infty \right)$ and decreasing elsewhere it is defined.

Again,

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

$\underset{x\to 0^{+}}{\lim }f\left(x\right)=\infty$ So, there is a vertical asymptote at $x=0$.