Question

To sketch the graph of the function which satisfy the following conditions that \({f^\prime }(0) = {f^\prime }(2) = {f^\prime }(4) = 0\;,\;{f^\prime }(x) > 0\) if \(x < 0\) or \(2 < x < 4\;,\;{f^\prime }(x) < 0\) if \(0 < x < 2\) or \(x > 4\;,\;{f^{\prime \prime }}(x) > 0\) if \(1 < x < 3\;,\;{f^{\prime \prime }}(x) < 0\) if \(x < 1\) or \(x > 3\).

Short answer

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

Step-by-step solution

Given data

The graph of the function which satisfy the following conditions are \({f^\prime }(0) = {f^\prime }(2) = {f^\prime }(4) = 0\), \({f^\prime }(x) > 0\) if \(x < 0\) or \(2 < x < 4,{f^\prime }(x) < 0\) if \(0 < x < 2\) or \(x > 4,{f^{\prime \prime }}(x) > 0\) if \(1 < x < 3,{f^{\prime \prime }}(x) < 0\) if \(x < 1\) or \(x > 3\).

Concept of Increasing test / decreasing test and concavity test

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".

Concavity Test:

If\({f^{\prime \prime }}(x) > 0\)for all\(x\)in\(I\), then the graph of\(f\)is concave upward on\({f^{\prime \prime }}(x) < 0\)for all\(x\)in\(I\), then the graph of\(f\)is concave downward on\({I^{\prime \prime }}\).

Sketch of the graph which satisfy the given conditions

The conditions \({f^\prime }(0) = {f^\prime }(2) = {f^\prime }(4) = 0\) means that horizontal tangents exists at \(x = 0\;,\;2\;,\;4\).

From the above definition the condition \({f^\prime }(x) > 0\) if \(x < 0\) or \(2 < x < 4\) indicates that \(f\) is increasing on the intervals \(( - \infty ,0)\) and \((2,4)\) and \({f^\prime }(x) < 0\) if \(0 < x < 2\) or \(x > 4\) indicates that \(f\) is decreasing on the intervals \((0,2)\) and \((4,\infty )\).

Similarly, \({f^{\prime \prime }}(x) > 0\) if \(1 < x < 3\) indicates that \(f\) is concave upward on \((1,3)\) and \({f^{\prime \prime }}(x) < 0\) if \(x < 1\) or \(x > 3\) indicates that \(f\) is concave downward on \(( - \infty ,1)\).

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

Notice that the Figure satisfies all the given conditions.