The following five graphs are, in some order, plots of , and ; that is, three unknown functions and the derivatives of the first two of those functions. Which graph is a plot of ?
Option (a) looks like a quartic, options (b), (d), and (e) look like cubics, and option (c) looks like a quadratic, so we might expect that one of the cubics is the derivative of option (a), and option (c) is the derivative of another of the cubics, leaving one remaining cubic to be . Option (c) has the right sign, and zeros in the right places, to be the derivative of option (b). Also, option (e) has the right number of zeros to be the derivative of option (a). That leaves option (d) as a possible candidate for . We should check that it's definitely not the derivative of any of the other options, and that it's derivative is not any of the other options. It's not the derivative of (a) or (c) or (e) because it's got the wrong number of zeros, and it's the not the derivative of (c) because it's negative for large . It's also not the case that the derivative of option (d) is any of the other graphs; such a graph would have two zeros for the two turning points of (d), but only (c) has two zeros, and the sign of option (c) is wrong (positive where the gradient of option (d) is negative). So we conclude that (c) is the derivative of (b), (e) is the derivative of (a) and option (d) is the "odd one out"; it is .