Chapter 3, Problem 41RE

Solution

To calculate: graph the function and analyse it.

Exact coordinates of absolute minimum point $\left(\pm \frac{1}{\sqrt{e}},-\frac{1}{2e}\right)$ .

Given information: The function is $f(x)=x^2\ln \left|x\right|$

Calculation:

The graph $f\left(x\right)$ as indicated:

  

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

Domain: all real $x\ne 0$ .

Range: all real $y\ge -0.184$ .

Continuity: continuous on domain.

Increasing or decreasing behaviour: decreasing $\left(-\infty ,-0.607\right)$ , increasing on $\left(0.607,\infty \right)$ .

Symmetry: the function is even:

  $\begin{array}{c}$ f(-x)={(-x)}^2\ln \left|-x\right|$=x^2\ln \left|x\right|\\ =f(x)\end{array}$

Boundedness: bounded below at $y=-0.184$ .

Extrema: $\left(\pm \mathrm{0.607.}-0.184\right)$

Asymptotes: none

End behaviour:

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

Exact coordinates of absolute minimum point found are

  $\left(\pm \frac{1}{\sqrt{e}},-\frac{1}{2e}\right)$

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