The First Derivative and the Shape of a Function f(x)

Practice 1: \(\mathrm{g}\) is increasing on \([2,4]\) and \([6,8]\).
\(\mathrm{g}\) is decreasing on \([0,2]\) and \([4,5]\),
\(g\) is constant on \([5,6]\).

Practice 2: The graph in Fig. 33 shows the rate of population change, \(\mathrm{dR} / \mathrm{dt}\).

Fig. 33

Practice 3: The graph of \(\mathrm{f}'\) is shown in Fig. 34. Notice how the graph of \(\mathrm{f}^{\prime}\) is \(0\) where \(\mathrm{f}\) has a maximum and minimum.

Fig. 34

Practice 4: The Second Shape Theorem for helicopters:
(i) If the upward velocity \(\mathrm{h}'\) is positive during time interval \(I\) then the height \(\mathrm{h}\) is increasing during time interval \(I\).
(ii) If the upward velocity \(\mathrm{h}'\) is negative during time interval \(I\) then the height \(\mathrm{h}\) is decreasing during time interval \(I\).
(iii) If the upward velocity \(\mathrm{h}'\) is zero during time interval \(I\) then the height \(\mathrm{h}\) is constant during time interval \(I\).

Practice 5: A graph satisfying the conditions in the table is shown in Fig. 35.

\(\begin{array}{l|r|r|l|l|l|l}
\mathrm{x} & -2 & -1 & 0 & 1 & 2 & 3 \\
\hline \mathrm{f}(\mathrm{x}) & 1 & -1 & -2 & -1 & 0 & 2 \\
\hline \mathrm{f}^{\prime}(\mathrm{x}) & -1 & 0 & 1 & 2 & -1 & 1
\end{array}\)

Fig. 35

Practice 6: \(\mathrm{f}(\mathrm{x})=\mathrm{x}^{3}-3 \mathrm{x}^{2}-24 \mathrm{x}+5\).

\(\mathrm{f}^{\prime}(\mathrm{x})=3 \mathrm{x}^{2}-6 \mathrm{x}-24=3(\mathrm{x}-4)(\mathrm{x}+2)\).

\(\mathrm{f}^{\prime}(\mathrm{x})=0\) if \(\mathrm{x}=-2,4\).

If \(x < -2\)
then \(f^{\prime}(x)=3(x-4)(x+2)=3\) (negative) (negative) \(> 0\) so \(f\) is increasing.

If \(-2 < x < 4\)
then \(\mathrm{f}^{\prime}(\mathrm{x})=3(\mathrm{x}-4)(\mathrm{x}+2)=3\) (negative)(positive) \(< 0\) so \(\mathrm{f}\) is decreasing.

If \(x > 4\)
then \(\mathrm{f}^{\prime}(\mathrm{x})=3(\mathrm{x}-4)(\mathrm{x}+2)=3\) (positive) (positive) \(> 0\) so \(\mathrm{f}\) is increasing

\(f\) has a relative maximum at \(x=-2\) and a relative minimum at \(x=4\).
The graph of \(\mathrm{f}\) is shown in Fig. 36.

Fig. 36

Practice 7: Fig. 37 shows several possible graphs for \(\mathrm{g}\). Each of the \(\mathrm{g}\) graphs has the correct shape to give the graph of \(\mathrm{g}'\). Notice that the graphs of \(\mathrm{g}\) are "parallel," differ by a constant.

Fig. 37

Practice 8: Fig. 38 shows the height graph for the balloon. The balloon was highest at \(4 \mathrm{pm}\) and had a local minimum at \(6 \mathrm{pm}\).

Fig. 38

Practice 9:
\(f(x)=3 x^{2}-12 x+7\) so \(f^{\prime}(x)=6 x-12\).

\(f^{\prime}(x)=0\) if \(x=2\).
If \(x < 2\), then \(f^{\prime}(x) < 0\) and \(f\) is decreasing.
If \(x > 2\), then \(f^{\prime}(x) > 0\) and \(f\) is increasing.
From this we can conclude that \(\mathrm{f}\) has a minimum when \(\mathrm{x}=2\) and has a shape similar to Fig. 19(b).

We could also notice that the graph of the quadratic \(f(x)=3 x^{2}-12 x+7\) is an upward opening parabola. The graph of \(\mathrm{f}\) is shown in Fig. 39.

Fig. 39