Difference between revisions of "2000 AMC 12 Problems/Problem 22"
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== Solution == | == 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, <math>P(-1) > 4</math> | # According to the graph, <math>P(-1) > 4</math> | ||
# The product of the roots is <math>d</math> by [[Vieta’s formulas]]. Also, <math>d = P(0) > 5</math> according to the graph. | # The product of the roots is <math>d</math> by [[Vieta’s formulas]]. Also, <math>d = P(0) > 5</math> according to the graph. | ||
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Clearly <math>\mathrm{(C)}</math> is the smallest. | Clearly <math>\mathrm{(C)}</math> is the smallest. | ||
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== See also == | == See also == | ||
{{AMC12 box|year=2000|num-b=21|num-a=23}} | {{AMC12 box|year=2000|num-b=21|num-a=23}} | ||
[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] |
Revision as of 13:13, 9 February 2009
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 .
Clearly is the smallest.
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 |