Question

Sketch the graph of the function which satisfy the following conditions that

vertical asymptote \(x = 0,\;{f^\prime }(x) > 0\) if \(x < - 2,\;{f^\prime }(x) < 0\) if \(x > - 2(x \ne 0),\;{f^{\prime \prime }}(x) < 0\) if \(x < 0,\;{f^{\prime \prime }}(x) > 0\) if \(x > 0\).

Short answer

The sketch of the function that satisfies the given conditions is shown below in figure 1.

Step-by-step solution

Given data

The vertical asymptote \(x = 0,\;{f^\prime }(x) > 0\) if \(x < - 2,\;{f^\prime }(x) < 0\) if \(x > - 2(x \ne 0),\;{f^{\prime \prime }}(x) < 0\) if \(x < 0,\;{f^{\prime \prime }}(x) > 0\) if \(x > 0\).

Concept of concavity test and increasing /decreasing test

Concavity test:

(a) If\({f^{\prime \prime }}(x) > 0\)for all\(x\)in\(l\), then\(f\)is concave upward on\(l\).

(b) If\({f^{\prime \prime }}(x) < 0\)for all\(x\)in\(l\), then\(f\)is concave downward on\(l\).

Increasing/decreasing test:

(a) If\({f^\prime }(x) > 0\)on an interval, then\(f\)is increasing on that interval.

(b) If\({f^\prime }(x) < 0\)on an interval, then\(f\)is decreasing on that interval.

Sketch of the function that satisfies the given conditions 

Vertical asymptote \(x = 0\) means that the graph of \(f(x)\) approaches infinity from either side near the point at \(x = 0\) but not exactly at \(x = 0\).

The condition \({f^\prime }(x) > 0\) if \(x < - 2\) indicates that \(f\) is increasing on \(( - \infty , - 2)\).

Similarly, \({f^\prime }(x) < 0\) if \(x > - 2\quad (x \ne 0)\) indicates \(f\) is decreasing on \(( - 2,0),{f^{\prime \prime }}(x) < 0\) if \(x < 0\) denotes that \(f\) is concave downward on \(( - \infty ,0)\) and \({f^{\prime \prime }}(x) > 0\) if \(x > 0\) denotes \(f\) is concave upward on \((0,\infty )\).

The rough sketch which satisfies the given conditions is shown below in figure 1.

Figure 1

Notice that the figure 1 satisfies all the given conditions.