Question

The function \(f\) is differentiable on the interval [-1,1] . The table shows the values of \(f^{\prime}\) for selected values of \(x\). Sketch the graph of \(f\), approximate the critical numbers, and identify the relative extrema. $$\begin{array}{|l|c|c|c|c|} \hline x & -1 & -0.75 & -0.50 & -0.25 \\ \hline f^{\prime}(x) & -10 & -3.2 & -0.5 & 0.8 \\ \hline \end{array}$$ $$\begin{array}{|l|l|l|l|l|l|} \hline \boldsymbol{x} & 0 & 0.25 & 0.50 & 0.75 & 1 \\ \hline \boldsymbol{f}^{\prime}(\boldsymbol{x}) & 5.6 & 3.6 & -0.2 & -6.7 & -20.1 \\ \hline \end{array}$$

Short answer

The critical numbers are \(x = -0.75\), \(x = -0.25\), \(x = 0\), \(x = 0.5\), and \(x = 0.75\). The function \(f\) has relative minimums at \(x = -0.75\) and \(x = 0.5\) and relative maximums at \(x = -0.25\), \(x = 0\), and \(x = 0.75\). The sketch of the function should reflect these observations.

Step-by-step solution

Identify sign changes in the derivative

Examine the function \(f^{\prime}\) values at given \(x\) points. The function \(f\) changes its increasing/decreasing behavior at the points where \(f^{\prime}\) changes sign. From the given tables we can see that the derivative changes sign at points \(x = -0.75\), \(x = -0.25\), \(x = 0\), \(x = 0.5\) and \(x = 0.75\). Hence these are the critical numbers.

Identify relative extrema

Take each of the critical numbers and check the sign of derivative before and after the point. If \(f^{\prime}\) changes from negative to positive, then \(f\) has a relative minimum there; if \(f^{\prime}\) changes from positive to negative, \(f\) has a relative maximum. \nFrom the tables, it can be observed that at \(x = -0.75\) and \(x = 0.5\), \(f^{\prime}\) changes sign from negative to positive. Hence, \(f\) has relative minimums at these points. Similarly, at \(x = -0.25\), \(x = 0\) and \(x = 0.75\), \(f^{\prime}\) changes sign from positive to negative indicating relative maximums at these points.

Sketch the graph

Start by plotting the critical points mentioned earlier. After that, remember that between two critical numbers where there are no sign changes in the derivative, the function is either entirely increasing or decreasing, as indicated by the positive or negative sign of the derivative, respectively. The sketch of function \(f\) should accordingly show increasing or decreasing behavior in those intervals. It should also show relative maximums and minimums at the points identified in Step 2.