Chapter 3, Problem 39RE

Solution

To graph: The function and analyse it.

Domain: all real $x\text{>}0$ .

Range: all real $y\ge -0.368$ .

Continuity: continuous on a domain.

Increasing or decreasing behaviour: decreasing $\left(0,0.368\right)$ , increasing on $\left(0.368,\infty \right)$ .

Symmetry: none.

Boundedness: bounded below at $y=-0.368$ .

Extrema: $\left(0.368-0.368\right)$

Asymptotes: none

End behaviour:

  $\begin{array}{c}\underset{x\to \infty }{\lim }f(x)=\infty \\ \underset{x\to 0^{+}}{\lim }f(x)=0\end{array}$

Exact coordinates of absolute minimum point found are

  $\left(\frac{1}{e},-\frac{1}{e}\right)$

Given information: The function is $f(x)=x\ln x$

Calculation:

The graph $f(x)$ is indicated below.

  

Now we should analyse the function based on the observation of graph:

Domain: all real $x\text{>}0$ .

Range: all real $y\ge -0.368$ .

Continuity: continuous on a domain.

Increasing or decreasing behaviour: decreasing $\left(0,0.368\right)$ , increasing on $\left(0.368,\infty \right)$ .

Symmetry: none.

Boundedness: bounded below at $y=-0.368$ .

Extrema: $\left(0.368-0.368\right)$

Asymptotes: none

End behaviour:

  $\begin{array}{c}\underset{x\to \infty }{\lim }f(x)=\infty \\ \underset{x\to 0^{+}}{\lim }f(x)=0\end{array}$

Exact coordinates of absolute minimum point found are

  $\left(\frac{1}{e},-\frac{1}{e}\right)$

But this cannot be proven without calculus, so we use decimal approximation as found in graphing utility.