Question

Describe how the graph of f varies as c varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when c changes. You should also identify any transitional values of c at which the basic shape of the curve changes.

33. \(f\left( x \right) = \frac{{cx}}{{1 + {c^2}{x^2}}}\)

Short answer

The points \(\left( {0,0} \right)\) and \(\left( { \pm \frac{{\sqrt 3 }}{c}, \pm \frac{{\sqrt 3 }}{4}} \right)\) is the point of inflection. Moreover, the coordinates of the inflection point are unrelated of \(c\). However, with \(c\) increases, both the inflection points approach the \(y\)-axis.

Step-by-step solution

Describe how the graph of f as c varies

It is observed that the transitional value is\(c = 0\)wherein the graph is about the\(x - \)axis.

Moreover, when we substitute\( - c\)for\(c\), then the function\(f\left( x \right) = \frac{{cx}}{{1 + {c^2}{x^2}}}\)would be reflected on the\(x\)-axis.

Therefore, we only examine at positive values of\(c\)(excepts\(c = - 1\), which is used to show the reflective property). Moreover,\(f\)is the odd function.

Since \(\mathop {\lim }\limits_{x \to \pm \infty } f\left( x \right) = 0\), therefore, the line \(y = 0\) is the horizontal asymptote for every \(c\).

Examine the maximum and minimum point and inflection point

Obtain the derivative of the function as shown below:

\(\begin{array}{c}f'\left( x \right) = \frac{d}{{dx}}\left( {\frac{{cx}}{{1 + {c^2}{x^2}}}} \right)\\ = \frac{{\left( {1 + {c^2}{x^2}} \right)c - cx\left( {2{c^2}x} \right)}}{{{{\left( {1 + {c^2}{x^2}} \right)}^2}}}\\ = - \frac{{c\left( {{c^2}{x^2} - 1} \right)}}{{{{\left( {1 + {c^2}{x^2}} \right)}^2}}}\end{array}\)

Take\(f'\left( x \right) = 0\)to obtain as shown below:

\(\begin{array}{c}f'\left( x \right) = 0\\{c^2}{x^2} - 1 = 0\\{x^2} = \frac{1}{{{c^2}}}\\x = \pm \frac{1}{c}\end{array}\)

According to the first derivative test, the absolute minimum value is\(f\left( {\frac{1}{c}} \right) = \frac{1}{2}\), and the absolute minimum value is\(f\left( { - \frac{1}{c}} \right) = - \frac{1}{2}\).

These extrema contain the same value regardless of\(c\). However, the maximum points become closer to the\(y\)-axis with\(c\)increases.

Obtain the second derivative of the function as shown below:

\(\begin{array}{c}f''\left( x \right) = \frac{d}{{dx}}\left( { - \frac{{c\left( {{c^2}{x^2} - 1} \right)}}{{{{\left( {1 + {c^2}{x^2}} \right)}^2}}}} \right)\\ = \frac{{\left( { - 2{c^3}x} \right){{\left( {1 + {c^2}{x^2}} \right)}^2} - \left( { - {c^3}{x^2} + c} \right)\left( {2\left( {1 + {c^2}{x^2}} \right)\left( {2{c^2}x} \right)} \right)}}{{{{\left( {1 + {c^2}{x^2}} \right)}^4}}}\\\frac{{\left( {1 + {c^2}{x^2}} \right)\left( {\left( { - 2{c^3}x} \right)\left( {1 + {c^2}{x^2}} \right) - \left( { - {c^3}{x^2} + c} \right)\left( {4{c^2}x} \right)} \right)}}{{{{\left( {1 + {c^2}{x^2}} \right)}^4}}}\\ = \frac{{\left( { - 2{c^3}x} \right)\left( {1 + {c^2}{x^2}} \right) + \left( {{c^3}{x^2} - c} \right)\left( {4{c^2}x} \right)}}{{{{\left( {1 + {c^2}{x^2}} \right)}^3}}}\\ = \frac{{2{c^3}x\left( {{c^2}{x^2} - 3} \right)}}{{{{\left( {1 + {c^2}{x^2}} \right)}^3}}}\end{array}\)

Take\(f''\left( x \right) = 0\)to obtain as shown below:

\(\begin{array}{c}\frac{{2{c^3}x\left( {{c^2}{x^2} - 3} \right)}}{{{{\left( {1 + {c^2}{x^2}} \right)}^3}}} = 0\\2{c^3}x\left( {{c^2}{x^2} - 3} \right) = 0\\{c^2}{x^2} - 3 = 0\\{x^2} = \frac{3}{{{c^2}}}\\x = \pm \frac{{\sqrt 3 }}{c},0\end{array}\)

The points\(\left( {0,0} \right)\)and\(\left( { \pm \frac{{\sqrt 3 }}{c}, \pm \frac{{\sqrt 3 }}{4}} \right)\)is the point of inflection.

Moreover, the coordinates of the inflection point are unrelated of\(c\), however, with\(c\)increases, both the inflection points reach the\(y\)-axis.

Graph of several members of the family

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

1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\(f\left( x \right) = \frac{{cx}}{{1 + {c^2}{x^2}}}\)in the\({Y_1}\)tab.
2. Set the values of\(c\)as\(0.2,0.5,1,2,4,5, - 1\)and observe the graph of the function.
3. Enter the “GRAPH” button in the graphing calculator.

Visualization of the graph of the function as shown below: