2018 AMC 12A Problems/Problem 21
Contents
Problem
Which of the following polynomials has the greatest real root?
Solution 1
We can see that our real solution has to lie in the open interval . From there, note that if , are odd positive integers if , so hence it can only either be B or E(as all of the other polynomials will be larger than the polynomial B). E gives the solution . We can approximate the root for B by using . therefore the root for B is approximately . The answer is . (cpma213)
Solution 2 (Calculus version of solution 1)
Note that and . Calculating the definite integral for each function on the interval , we see that gives the most negative value. To maximize our real root, we want to maximize the area of the curve under the x-axis, which means we want our integral to be as negative as possible and thus the answer is .
Solution 3 (Alternate Calculus Version)
Newton's Method is used to approximate the zero of any real valued function given an estimation for the root : After looking at all the options, gives a reasonable estimate. For options A to D, and the estimation becomes Thus we need to minimize the derivative, giving us B. Now after comparing B and E through Newton's method, we see that B has the higher root, so the answer is . (Qcumber)
Video Solution by Richard Rusczyk
https://artofproblemsolving.com/videos/amc/2018amc12a/471
~ dolphin7
See Also
2018 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 20 |
Followed by Problem 22 |
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.