Chapter 4.4, Problem 9E

To determine

To Find:The intervals of the increase and decrease and intervals concavity of the function.

Answer to Problem 9E

Increasing on $(-8-\sqrt{61},-8+\sqrt{61})$

Decreasing on $(-\infty ,-8-\sqrt{61}),(-8+\sqrt{61},0),(0,\infty )$

Concave up $(-12-\sqrt{138},-12+\sqrt{138}),(0,\infty )$

Concave down on $(-\infty ,-12-\sqrt{138}),(-12+\sqrt{138},0)$

Explanation of Solution

Given:Produce graph of $f$ that reveals all the aspects of the curve. Estimate the intervals of the increase and decrease and intervals concavity and use the calculus to find these intervals exactly.

  $f(x)=1+\frac{1}{x}+\frac{8}{x^2}+\frac{1}{x^3}$

Given: $f(x)=1+\frac{1}{x}+\frac{8}{x^2}+\frac{1}{x^3}$

The graph of the function is

  

From the graph it is concluded that

The asymptotes is $y=1,x=0$

The spread out image of the graph is shown below

  

So,

The local maxima is around $x=-0.17$ and an inflection point is $x=-0.25$

Thus,

The function is

Increasing on $(-\infty ,-0.17)$

Decreasing on $(-0.17,0),(0,\infty )$

Concave up $(-\infty ,-0.25),(0,\infty )$

Concave down on $(-0.25,0)$

Finding the exact values by differentiating

  $\begin{array}{l}y\text{'}=-\frac{1}{x^2}+\frac{-16}{x^3}+\frac{-3}{x^4}\\ =\frac{x^2+16x+3}{-x^4}\end{array}$

Finding the factors of numerator using the discriminant rule

  $\begin{array}{l}=x^2+16x+3\\ x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ x=\frac{-16\pm \sqrt{244}}{2}\end{array}$

We get

  $x\approx -15.810,-0.190$

It is indicated that the local maxima is $x\approx -15.810$

Another inflection point is $x=23$

Since the graph approaches $y=-1$

Thus

Modified intervals

Increasing on $(-8-\sqrt{61},-8+\sqrt{61})$

Decreasing on $(-\infty ,-8-\sqrt{61}),(-8+\sqrt{61},0)$

Concave up $(-23,-0.25)$

Concave down on $(-\infty ,-23),(-0.25,0)$

Now checking for $y^n$

  $\begin{array}{l}{y}^n=\frac{2}{x}+\frac{48}{x}+\frac{12}{x}\\ =\frac{2(x^2+24x+6)}{x^5}\end{array}$

Factors of the numerator are $x=-23.7,-0.253$

Hence the exact intervals are

Increasing on $(-8-\sqrt{61},-8+\sqrt{61})$

Decreasing on $(-\infty ,-8-\sqrt{61}),(-8+\sqrt{61},0),(0,\infty )$

Concave up $(-12-\sqrt{138},-12+\sqrt{138}),(0,\infty )$

Concave down on $(-\infty ,-12-\sqrt{138}),(-12+\sqrt{138},0)$