$$ \begin{array}{|c||c|c|c|c|c|} \hline x & -3 & -1 & 0 & 2 & 4 \\ \hline f'(x) & 5 & 2 & 5 &2 & 1 \\ \hline \end{array} $$

Let \(f\) be a polynomial function with values \(f'(x)\) at selected values of \(x\) given in the table above. Which of the following must be true for \(-3 \lt x \lt 4\) ?

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You may be tempted to say that the graph of \(f\) is increasing and has no critical points, but reflect that it is possible that there are points not shown in the table that are negative.

The graph of \(f\) would be concave up if values in the table were strictly increasing, but the values decrease, increase, and decrease.

Since \(f'(x)\) changes from decreasing to increasing somewhere from \(-3\lt x\lt 0\) and changes from increasing to decreasing somewhere from \(-1\lt x\lt 2\), the graph of \(f\) must have two inflection points minimum.