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Chapter 4: Q21E (page 279)

Use a computer algebra system to graph \(f\) and to find \(f'\) and \(f''\). Use graphs of these derivatives to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points of \(f\).

21. \(f\left( x \right) = \frac{{1 - {e^{{1 \mathord{\left/

{\vphantom {1 x}} \right.

\kern-\nulldelimiterspace} x}}}}}{{1 + {e^{{1 \mathord{\left/

{\vphantom {1 x}} \right.

\kern-\nulldelimiterspace} x}}}}}\)

Short Answer

Expert verified

\(f'\left( x \right) = \frac{{2{e^{{1 \mathord{\left/

{\vphantom {1 x}} \right.

\kern-\nulldelimiterspace} x}}}}}{{{x^2}{{\left( {1 + {e^{{1 \mathord{\left/

{\vphantom {1 x}} \right.

\kern-\nulldelimiterspace} x}}}} \right)}^2}}}\)

\(f''\left( x \right) = \frac{{ - 2{e^{{1 \mathord{\left/

{\vphantom {1 x}} \right.

\kern-\nulldelimiterspace} x}}}\left( {1 - {e^{{1 \mathord{\left/

{\vphantom {1 x}} \right.

\kern-\nulldelimiterspace} x}}} + 2x + 2x{e^{{1 \mathord{\left/

{\vphantom {1 x}} \right.

\kern-\nulldelimiterspace} x}}}} \right)}}{{{x^4}{{\left( {1 + {e^{{1 \mathord{\left/

{\vphantom {1 x}} \right.

\kern-\nulldelimiterspace} x}}}} \right)}^3}}}\)

Decreasing Interval: None

Increasing intervals: \(\left( { - \infty ,0} \right)\) and \(\left( {0,\infty } \right)\)

Local maximum value:None

Local minimum values:None

Concave up intervals: \(\left( { - \infty , - 0.417} \right)\), and \(\left( {0,0.417} \right)\)

Concave down intervals: \(\left( { - 0.417,0} \right)\) and \(\left( {0.417,\infty } \right)\)

Inflection points: \(\left( { - 0.417,0.834} \right)\), and \(\left( {0.417, - 0.834} \right)\)

Step by step solution

01

Determine the first derivative of the function

The given function is \(f\left( x \right) = \frac{{1 - {e^{{1 \mathord{\left/

{\vphantom {1 x}} \right.

\kern-\nulldelimiterspace} x}}}}}{{1 + {e^{{1 \mathord{\left/

{\vphantom {1 x}} \right.

\kern-\nulldelimiterspace} x}}}}}\).

Use the following steps to find the derivative of \(f\left( x \right)\) in Wolframalpha.com.

  1. Enter the function \(f\left( x \right) = \frac{{1 - {e^{{1 \mathord{\left/
  2. {\vphantom {1 x}} \right.
  3. \kern-\nulldelimiterspace} x}}}}}{{1 + {e^{{1 \mathord{\left/
  4. {\vphantom {1 x}} \right.
  5. \kern-\nulldelimiterspace} x}}}}}\)in the tab in the form of \(\frac{{\rm{d}}}{{{\rm{d}}x}}\left( {\frac{{1 - {e^{{1 \mathord{\left/
  6. {\vphantom {1 x}} \right.
  7. \kern-\nulldelimiterspace} x}}}}}{{1 + {e^{{1 \mathord{\left/
  8. {\vphantom {1 x}} \right.
  9. \kern-\nulldelimiterspace} x}}}}}} \right)\).
  10. Press the \( = \) button on the right side of the tab.

So, the derivative of \(f\left( x \right)\) is:

\(f'\left( x \right) = \frac{{2{e^{{1 \mathord{\left/

{\vphantom {1 x}} \right.

\kern-\nulldelimiterspace} x}}}}}{{{x^2}{{\left( {1 + {e^{{1 \mathord{\left/

{\vphantom {1 x}} \right.

\kern-\nulldelimiterspace} x}}}} \right)}^2}}}\)

02

Determine the second derivative of the function

Differentiate the function again using Wolfarmalpha.com calculator.

  1. Enter the function \(f'\left( x \right) = \frac{{2{e^{{1 \mathord{\left/
  2. {\vphantom {1 x}} \right.
  3. \kern-\nulldelimiterspace} x}}}}}{{{x^2}{{\left( {1 + {e^{{1 \mathord{\left/
  4. {\vphantom {1 x}} \right.
  5. \kern-\nulldelimiterspace} x}}}} \right)}^2}}}\)in the tab in the form of \(\frac{{\rm{d}}}{{{\rm{d}}x}}\left( {\frac{{2{e^{{1 \mathord{\left/
  6. {\vphantom {1 x}} \right.
  7. \kern-\nulldelimiterspace} x}}}}}{{{x^2}{{\left( {1 + {e^{{1 \mathord{\left/
  8. {\vphantom {1 x}} \right.
  9. \kern-\nulldelimiterspace} x}}}} \right)}^2}}}} \right)\).
  10. Press the \( = \) button on the right side of the tab.

So, the derivative of \(f'\left( x \right)\) is:

\(f''\left( x \right) = \frac{{ - 2{e^{{1 \mathord{\left/

{\vphantom {1 x}} \right.

\kern-\nulldelimiterspace} x}}}\left( {1 - {e^{{1 \mathord{\left/

{\vphantom {1 x}} \right.

\kern-\nulldelimiterspace} x}}} + 2x + 2x{e^{{1 \mathord{\left/

{\vphantom {1 x}} \right.

\kern-\nulldelimiterspace} x}}}} \right)}}{{{x^4}{{\left( {1 + {e^{{1 \mathord{\left/

{\vphantom {1 x}} \right.

\kern-\nulldelimiterspace} x}}}} \right)}^3}}}\)

03

Graphs of \(f\left( x \right)\), \(f'\left( x \right)\) and \(f''\left( x \right)\)

The procedure to draw the graph of the above equation by using the graphing calculator is as follows:

To check the answer, visually draw the graph of functions\(f\left( x \right) = \frac{{1 - {e^{{1 \mathord{\left/

{\vphantom {1 x}} \right.

\kern-\nulldelimiterspace} x}}}}}{{1 + {e^{{1 \mathord{\left/

{\vphantom {1 x}} \right.

\kern-\nulldelimiterspace} x}}}}}\),\(f'\left( x \right) = \frac{{2{e^{{1 \mathord{\left/

{\vphantom {1 x}} \right.

\kern-\nulldelimiterspace} x}}}}}{{{x^2}{{\left( {1 + {e^{{1 \mathord{\left/

{\vphantom {1 x}} \right.

\kern-\nulldelimiterspace} x}}}} \right)}^2}}}\)and\(f''\left( x \right) = \frac{{ - 2{e^{{1 \mathord{\left/

{\vphantom {1 x}} \right.

\kern-\nulldelimiterspace} x}}}\left( {1 - {e^{{1 \mathord{\left/

{\vphantom {1 x}} \right.

\kern-\nulldelimiterspace} x}}} + 2x + 2x{e^{{1 \mathord{\left/

{\vphantom {1 x}} \right.

\kern-\nulldelimiterspace} x}}}} \right)}}{{{x^4}{{\left( {1 + {e^{{1 \mathord{\left/

{\vphantom {1 x}} \right.

\kern-\nulldelimiterspace} x}}}} \right)}^3}}}\)by using the graphing calculator as shown below:

  1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\(\left( {1 - {e^{{1 \mathord{\left/
  2. {\vphantom {1 X}} \right.
  3. \kern-\nulldelimiterspace} X}}}} \right)/\left( {1 + {e^{{1 \mathord{\left/
  4. {\vphantom {1 X}} \right.
  5. \kern-\nulldelimiterspace} X}}}} \right)\)in\({Y_1}\)the taband\(2{e^{{1 \mathord{\left/
  6. {\vphantom {1 X}} \right.
  7. \kern-\nulldelimiterspace} X}}}/{X^2}{\left( {1 + {e^{{1 \mathord{\left/
  8. {\vphantom {1 X}} \right.
  9. \kern-\nulldelimiterspace} X}}}} \right)^2}\)in the\({Y_2}\)tab.
  10. Enter the “GRAPH” button in the graphing calculator.

Visualization of the graph of \(f\left( x \right)\) and \(f'\left( x \right)\)is shown below:

For\(f''\left( x \right)\):

  1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\( - 2{e^{{1 \mathord{\left/
  2. {\vphantom {1 X}} \right.
  3. \kern-\nulldelimiterspace} X}}}\left( {1 - {e^{{1 \mathord{\left/
  4. {\vphantom {1 X}} \right.
  5. \kern-\nulldelimiterspace} X}}} + 2X + 2X{e^{{1 \mathord{\left/
  6. {\vphantom {1 X}} \right.
  7. \kern-\nulldelimiterspace} X}}}} \right)/{X^4}{\left( {1 + {e^{{1 \mathord{\left/
  8. {\vphantom {1 X}} \right.
  9. \kern-\nulldelimiterspace} X}}}} \right)^3}\)in the\({Y_1}\)tab.
  10. Enter the “GRAPH” button in the graphing calculator.

Visualization of the graph of the function\(f''\left( x \right)\)is shown below:

04

Determine increase decrease intervals

From the graph of\(f\)and \(f'\), the domain of the function\(f\)is \(\left( { - \infty ,0} \right) \cup \left( {0,\infty } \right)\), there is no intercepts of \(f\).

So, the function is increasing on \(\left( { - \infty ,0} \right)\) and \(\left( {0,\infty } \right)\).

05

Determine local minimum and maximum values

There are no extreme values, as function is only increasing.

06

Determine intervals of concavity and inflection points

From the graph of\(f''\left( x \right)\), it can be concluded that there are two zeros, so there will be two inflection points. Where,\(f\)is concave upon \(\left( { - \infty , - 0.417} \right)\), and \(\left( {0,0.417} \right)\).

The function is concave down on \(\left( { - 0.417,0} \right)\) and \(\left( {0.417,\infty } \right)\).

And the inflection points from the graph are about, \(\left( { - 0.417,0.834} \right)\), and \(\left( {0.417, - 0.834} \right)\).

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Most popular questions from this chapter

(a) Sketch the graph of a function on \(\left( { - 1,2} \right)\) that has an absolute maximum but no local maximum.

(b) Sketch the graph of a function on \(\left( { - 1,2} \right)\) that has a local maximum but no absolute maximum.

(a) Sketch the graph of a function on \(\left( { - 1,2} \right)\) that has an absolute maximum but no absolute minimum.

(b) Sketch the graph of a function on \(\left( { - 1,2} \right)\) that is discontinuous but has both an absolute maximum and absolute minimum.

In Example 4 we considered a member of the family of functions \(f\left( x \right) = \sin \left( {x + \sin cx} \right)\) that occurs in FM synthesis. Here we investigate the function with \(c = 3\). Start by graphing f in the viewing rectangle \(\left( {0,\pi } \right)\) by \(\left( { - 1.2,1.2} \right)\). How many local maximum points do you see? The graph has more than are visible to the naked eye. To discover the hidden maximum and minimum points you will need to examine the graph of \(f'\) very carefully. In fact, it helps to look at the graph of \(f''\) at the same time. Find all the maximum and minimum values and inflection points. Then graph \(f\) in the viewing rectangle\(\left( { - 2\pi ,2\pi } \right)\) by \(\left( { - 1.2,1.2} \right)\) and comment on symmetry?

Use the guidelines of this section to sketch the curve. In guideline D, find an equation of the slant asymptote.

\(y = \frac{{{x^3} + 4}}{{{x^2}}}\)

65-68 Find an equation of Slant asymptote. Do not sketch the curve.

67. \(y = \frac{{{\bf{2}}{x^{\bf{3}}} - {\bf{5}}{x^{\bf{2}}} + {\bf{3}}x}}{{{x^{\bf{2}}} - x - {\bf{2}}}}\)
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