Chapter 3, Problem 63RE

To determine

To draw: The graph $f$.

Answer to Problem 63RE

Graph of $f$ from $f^{\prime }$.

  

Explanation of Solution

Given information:

  $f^{\prime }\left(x\right)=\{\begin{array}{l}-2,x<1\\ 1,\text{}\text{}\text{}\text{}\text{}\text{}1<x<4\\ -1,\text{}\text{}4<x<6\end{array}$ , $f\left(-1\right)=2$

From the given information we know the function $f\left(x\right)$ has a point $\left(1,2\right)$.

Let’s plot the point $\left(1,2\right)$.

The first derivative of $f\left(x\right)$ would give the slope of $f\left(x\right)$ . Hence draw a constant line through the point $\left(1,2\right)$ with the slope $-2$ . Then end the line on $x<1$.

Then change the direction of the line on $x<1$ with a slope of $1$ and reach $x<4$.

Then change the direction of the line on $x<4$ with a slope of $1$ and reach $x<6$.

Thus we can graph $f\left(x\right)$ of the given $f^{\prime }$.