Question

13-14 Sketch the graph by hand using asymptotes and intercepts, but not derivatives. Then use your sketch as a guide to producing graphs using a calculator or computer that displays the major features of the curve. Use these graphs to estimate the maximum and minimum values.

14. \(f\left( x \right) = \frac{{{{\left( {{\bf{2}}x + {\bf{3}}} \right)}^{\bf{2}}}\,{{\left( {x - {\bf{2}}} \right)}^{\bf{5}}}}}{{{x^{\bf{3}}}{{\left( {x - {\bf{5}}} \right)}^{\bf{2}}}}}\)

Short answer

The graph is shown below:

The local minimum value of \(f\left( {7.98} \right) \approx 609\) is the local minimum value.

Step-by-step solution

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

For the function \(f\left( x \right)\), \(x \to {0^ - }\), \(\mathop {\lim }\limits_{x \to {0^ - }} f\left( x \right) = \infty \) and \(x \to {0^ + }\), \(\mathop {\lim }\limits_{x \to {0^ + }} f\left( x \right) = \infty \). So, \(x = 0\) is a vertical asymptote. Similarly, \(x = 5\) is also a vertical asymptote.

As \(\mathop {\lim }\limits_{x \to \pm \infty } f\left( x \right) = \infty \), the curve of f does not have any horizontal asymptote.

As f is not defined for \(x = 0\), therefore no y-intercept.

The x-intercepts of the curve can be calculated by the roots of \(f\left( x \right) = 0\).

\(\begin{array}{c}{\left( {2x + 3} \right)^2}{\left( {x - 2} \right)^5} = 0\\x = - \frac{3}{2},2\end{array}\)

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

Find the maximum and minimum value of the function

The function is not defined at \(x = 0\), and there is no y-intercept. The only tangent of the graph is the x-axis, and it doesn’t cross it at \(x = - \frac{3}{2}\), since fis positive as \(x \to - {\left( {\frac{3}{2}} \right)^ - }\) and \(x \to {\left( {\frac{3}{2}} \right)^ + }\). There is a local minimum value of \(f\left( { - \frac{3}{2}} \right) = 0\).

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

1. In the graphing calculator, select “STAT PLOT” and enter the equations \(\frac{{{{\left( {2x + 3} \right)}^2}{{\left( {x - 2} \right)}^5}}}{{{x^3}{{\left( {x - 5} \right)}^2}}}\).
2. Set the window size \(5 \le X \le 10\), and \(0 \le Y \le 1000\).
3. Enter the graph button in the graphing calculator.

From the above graphs, it can be observed that the local minimum value of \(f\left( {7.98} \right) \approx 609\) is the local minimum value.