2000 AMC 12 Problems/Problem 22

Problem

The graph below shows a portion of the curve defined by the quartic polynomial $P(x) = x^4 + ax^3 + bx^2 + cx + d$ . Which of the following is the smallest?

Note that there are 3 maxima/minima. Hence we know that the rest of the graph is greater than 10. We approximate each of the above expressions:

According to the graph, $P(-1) > 4$
The product of the roots is $d$ by Vieta’s formulas. Also, $d = P(0) > 5$ according to the graph.
The product of the real roots is $>5$ , and the total product is $<6$ (from above), so the product of the non-real roots is $< \frac{6}{5}$ .
The sum of the coefficients is $P(1) > 1.5$
The sum of the real roots is $> 5$ .

Clearly $\mathrm{(C)}$ is the smallest.