Chapter 4, Problem 11RE

(a)

To determine

To find: The interval on which given function is increasing.

Answer to Problem 11RE

The given function increases in the interval $(0,2]$ .

Explanation of Solution

Given:

The function is $y=\ln \left|x\right|,-2\le x\le 2,x\ne 0$

Calculation:

The derivative of the function is:

When $x>0,y\text{'}=\frac{d}{dx}\ln x=\frac{1}{x}$

  $x<0,y\text{'}=\frac{d}{dx}\ln (-x)=-\frac{1}{x}(-1)=\frac{1}{x}$

Hence,

  $y\text{'}=\frac{1}{x}$ for all values of x in the domain.

Intervals	$(-2,0)$	$(0,2)$
Sign of y’	-	+
Nature	Decreasing	Increasing

Conclusion:

The given function increases in the interval $(0,2]$ .

(b)

To determine

To find: The interval on which given function is decreasing.

Answer to Problem 11RE

The given function increases in the interval $[-2,0)$ .

Explanation of Solution

Given:

The function is $y=\ln \left|x\right|,-2\le x\le 2,x\ne 0$

Calculation:

Intervals	$(-2,0)$	$(0,2)$
Sign of y’	-	+
Nature	Decreasing	Increasing

Conclusion:

The given function increases in the interval $[-2,0)$ .

(c)

To determine

The interval in which function is concave up.

Answer to Problem 11RE

The function is not concave up in any interval.

Explanation of Solution

Given:

The function is $y=\ln \left|x\right|,-2\le x\le 2,x\ne 0$

Calculation:

The second derivative of the function is:

  $y\text{'}\text{'}=-x^{-2}$

As, the second derivative is negative therefore, the function is not concave up. It is concave down.

Conclusion:

The function is not concave up in any interval.

(d)

To determine

The interval in which function is concave down.

Answer to Problem 11RE

The function is concave down in $(-2,0)$ and $(0,2)$ .

Explanation of Solution

Given:

The function is $y=\ln \left|x\right|,-2\le x\le 2,x\ne 0$

Calculation:

The second derivative of the function is:

  $y\text{'}\text{'}=-x^{-2}$

The function is concave down because the second derivative is negative.

The interval is between $(-2,0)$ and $(0,2)$ .

Conclusion:

The function is concave down in $(-2,0)$ and $(0,2)$ .

(e)

To determine

The local extreme values.

Answer to Problem 11RE

The local extreme and absolute maxima is at $(-2,\ln 2),(2,ln2)$

Explanation of Solution

Given:

The function is $y=\ln \left|x\right|,-2\le x\le 2,x\ne 0$

Calculation:

The interval is between $(-2,0)$ and $(0,2)$ .

Conclusion:

The local extreme and absolute maxima is at $(-2,\ln 2),(2,ln2)$

(f)

To determine

The inflection points.

Answer to Problem 11RE

There are no inflection points.

Explanation of Solution

Given:

The function is $y=\ln \left|x\right|,-2\le x\le 2,x\ne 0$

Calculation:

The function is concave down so, there is no inflection point.

Conclusion:

There are no inflection points.