Chapter 4.3, Problem 51E

To determine

The sketch of a possible graph of a continuous function $f$ that has the given properties.

Answer to Problem 51E

It has been determined that the sketch of a possible graph of a continuous function $f$ that has the given properties, is as follows:

  

Explanation of Solution

Given:

Domain $\left[0,6\right]$ , $f\left(0\right)=2$ and the following graph of $f^{\prime }$:

  

Concept used:

A point where $f^{\prime }=0$ is a point of local minima if $f^{\prime }$ is increasing at that point. Similarly, a point where $f^{\prime }=0$ is a point of local maxima if $f^{\prime }$ is decreasing at that point. A point of discontinuity in the graph of $f^{\prime }$ corresponds to a sharp turn in the graph of $f$ .

Calculation:

It is given that the continuous function $f$ has a domain of $\left[0,6\right]$ such that $f\left(0\right)=2$ and the graph of its first derivative is as follows:

  

Then, a possible sketch of the graph of $f$ is as follows:

  

Conclusion:

It has been determined that the sketch of a possible graph of a continuous function $f$ that has the given properties, is as follows: