Question

Sketch the graph of the function that satisfies all of the given conditions.

38. \(f'\left( x \right) > {\bf{0}}\)for \(x \ne {\bf{1}}\). Vertical asymptote \(x = {\bf{1}}\).

\(f''\left( x \right) > {\bf{0}}\)if \[x < {\bf{1}}\] or \[x > {\bf{3}}\], \(f''\left( x \right) < {\bf{0}}\) if \[{\bf{1}} < x < {\bf{3}}\].

Short answer

The required graph is given below.

Step-by-step solution

Write an interpretation of the given information about f

The following observations can be made about f from the given information:

1. As\(f'\left( x \right) > 0\)for\(x \ne 1\), f is increasing in the intervals\(\left( { - \infty ,1} \right)\)and\(\left( {1,\infty } \right)\).
2. As\(f''\left( x \right) > 0\)for\(x < 1\)or\(x > 3\), f is concave upward in\(\left( { - \infty ,1} \right)\)and\(\left( {3,\infty } \right)\).
3. As\(f''\left( x \right) < 0\)for\(1 < x < 3\), f is concave downward in\(\left( {1,3} \right)\).

There is an inflection point at \(x = 3\).

Sketch the possible graph of \(f\left( x \right)\)

By using the nature of the curve given in Step 1, plot the sketch of the curve of \(f\left( x \right)\).

The figure given below represents the possible curve of \(f\left( x \right)\).