Question

9-10 Produce graphs of f that reveal all the important aspects of the curve. Estimate the intervals of increase and decrease and intervals of concavity, and use calculus to find these intervals exactly.

10. \(f\left( x \right) = \frac{{\bf{1}}}{{{x^{\bf{8}}}}} - \frac{{{\bf{2}} \times {\bf{1}}{{\bf{0}}^{\bf{8}}}}}{{{x^{\bf{4}}}}}\)

Short answer

The graph is shown below:

The function f increases on \(\left( { - 0.01,0} \right)\) and \(\left( {0.01,\infty } \right)\) and decreases on \(\left( { - \infty , - 0.01} \right)\) and \(\left( {0,0.01} \right)\). Also, f has a local minimum value at \(x = \pm 0.01\) and the minimum value is \( - {10^{16}}\).

The curve of f is concave upward on \(\left( { - 0.012,0} \right)\) and \(\left( {0,0.012} \right)\) and is concave downward on \(\left( { - \infty , - 0.012} \right)\) and \(\left( {0.012,\infty } \right)\).

The curve of the function f has concave upward on \(\left( { - \frac{1}{{100}}\sqrt(4){{1.8}},0} \right)\) and \(\left( {0, - \frac{1}{{100}}\sqrt(4){{1.8}}} \right)\) and \(f''\) is concave downward on \(\left( { - \infty , - \frac{1}{{100}}\sqrt(4){{1.8}}} \right)\) and \(\left( {\frac{1}{{100}}\sqrt(4){{1.8}},\infty } \right)\).

Step-by-step solution

Differentiate the function \(f\left( x \right)\)

Differentiate the function \(f\left( x \right) = \frac{1}{{{x^8}}} - \frac{{2 \times {{10}^8}}}{{{x^4}}}\).

\(\begin{array}{c}f'\left( x \right) = \frac{{\rm{d}}}{{{\rm{d}}x}}\left( {\frac{1}{{{x^8}}} - \frac{{2 \times {{10}^8}}}{{{x^4}}}} \right)\\ = - \frac{8}{{{x^9}}} + \frac{{8 \times {{10}^8}}}{{{x^5}}}\end{array}\)

Differentiate the function \(f'\left( x \right) = - \frac{8}{{{x^9}}} + \frac{{8 \times {{10}^8}}}{{{x^5}}}\).

\(\begin{array}{c}f''\left( x \right) = \frac{{\rm{d}}}{{{\rm{d}}x}}\left( { - \frac{8}{{{x^9}}} + \frac{{8 \times {{10}^8}}}{{{x^5}}}} \right)\\ = \frac{{72}}{{{x^{10}}}} - \frac{{40 \times {{10}^8}}}{{{x^6}}}\end{array}\)

Sketch the graph of f, \(f'\), and \(f''\)

Use the following steps to plot the graph of the given functions:

1. In the graphing calculator, select “STAT PLOT” and enter the equation \(\frac{1}{{{x^8}}} - \frac{{2 \times {{10}^8}}}{{{x^4}}}\) in the \({Y_1}\) tab.
2. Enter the graph button in the graphing calculator.

The figure below represents the graph of \(f\left( x \right)\).

Write an interpretation of graphs of f

It appears that f increases on \(\left( { - 0.01,0} \right)\) and \(\left( {0.01,\infty } \right)\) and decrease on \(\left( { - \infty , - 0.01} \right)\) and \(\left( {0,0.01} \right)\), that f has a local minimum value at \(x = \pm 0.01\) and the minimum value is \( - {10^{16}}\).

The curve of f is concave upward on \(\left( { - 0.012,0} \right)\) and \(\left( {0,0.012} \right)\) and is concave downward on \(\left( { - \infty , - 0.012} \right)\) and \(\left( {0.012,\infty } \right)\).

Find the exact value intervals

The root of the equation \(f'\left( x \right) = 0\) can be calculated as follows:

\(\begin{array}{c} - \frac{8}{{{x^9}}} + \frac{{8 \times {{10}^8}}}{{{x^5}}} = 0\\{x^4} = \sqrt {\frac{1}{{{{10}^8}}}} \\ = \pm \frac{1}{{100}}\end{array}\)

So, \(f'\left( x \right) > 0\) on \(\left( { - 0.01,0} \right)\) and \(\left( {0.01,\infty } \right)\) and \(f'\left( x \right) < 0\) on \(\left( { - \infty , - 0.01} \right)\) and \(\left( {0,0.01} \right)\).

Find the root of the equation \(f''\left( x \right) = 0\) that can be calculated as follows:

\(\begin{array}{c}\frac{{72}}{{{x^{10}}}} - \frac{{40 \times {{10}^8}}}{{{x^6}}} = 0\\{x^4} = \frac{9}{{5 \times {{10}^8}}}\\ = \pm \frac{3}{{\sqrt(4){5} \times 100}}\\ = \pm 0.0116\end{array}\)

So, the curve of f has concave upward on \(\left( { - \frac{1}{{100}}\sqrt(4){{1.8}},0} \right)\) and \(\left( {0, - \frac{1}{{100}}\sqrt(4){{1.8}}} \right)\) and \(f''\) is concave downward on \(\left( { - \infty , - \frac{1}{{100}}\sqrt(4){{1.8}}} \right)\) and \(\left( {\frac{1}{{100}}\sqrt(4){{1.8}},\infty } \right)\).