Difference between revisions of "2000 AMC 12 Problems/Problem 22"
Fnothing4994 (talk | contribs) (→Solution) |
(→Solution) |
||
| Line 19: | Line 19: | ||
Clearly <math>\mathrm{(C)}</math> is the smallest. | Clearly <math>\mathrm{(C)}</math> is the smallest. | ||
| + | |||
| + | == Video Solution == | ||
| + | https://youtu.be/MMIEbkGu-k8 | ||
== See Also == | == See Also == | ||
Revision as of 22:26, 24 October 2021
Contents
Problem
The graph below shows a portion of the curve defined by the quartic polynomial 
. Which of the following is the smallest?
Solution
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,
- The product of the roots is
by Vieta’s formulas. Also, 
according to the graph. - The product of the real roots is
, and the total product is 
(from above), so the product of the non-real roots is 
. - The sum of the coefficients is
- The sum of the real roots is
.
by 
according to the graph.
, and the total product is 
(from above), so the product of the non-real roots is 
.
.Clearly 
is the smallest.
Video Solution
See Also
| 2000 AMC 12 (Problems • Answer Key • Resources) | |
| Preceded by Problem 21 |
Followed by Problem 23 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
