Question

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

36. Vertical asymptote \(x = {\bf{0}}\), \(f'\left( x \right) > {\bf{0}}\)if \(x < - {\bf{2}}\), and \(f''\left( x \right) < {\bf{0}}\), for all \(x\)

\(f'\left( x \right) < {\bf{0}}\)if \(x > - {\bf{2}}\)\(\left( {x = {\bf{0}}} \right)\), \(f''\left( x \right) < {\bf{0}}\)if \[x < {\bf{0}}\], \(f''\left( x \right) > {\bf{0}}\) if \[x > {\bf{0}}\].

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\(f'\left( x \right) > 0\)and\(x < - 2\), it shows that f is increasing in the interval\(\left( { - \infty , - 2} \right)\).
2. As\(f'\left( x \right) < 0\)and\(x > - 2\), it shows that f is decreasing in the intervals\[\left( { - 2,0} \right)\]and\(\left( {0,\infty } \right)\).
3. As\(f''\left( x \right) < 0\)for\(x < 0\), f is concave downward in\(\left( { - \infty ,0} \right)\).
4. As\(f''\left( x \right) > 0\)for\(x > 0\), f is concave upward in \(\left( {0,\infty } \right)\).

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 \(f\left( x \right)\).