Difference between revisions of "2018 AMC 12A Problems/Problem 21"

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<math>\textbf{(A) }  x^{19}+2018x^{11}+1  \qquad        \textbf{(B) }  x^{17}+2018x^{11}+1  \qquad    \textbf{(C) }  x^{19}+2018x^{13}+1  \qquad  \textbf{(D) }  x^{17}+2018x^{13}+1 \qquad  \textbf{(E) }  2019x+2018 </math>
 
<math>\textbf{(A) }  x^{19}+2018x^{11}+1  \qquad        \textbf{(B) }  x^{17}+2018x^{11}+1  \qquad    \textbf{(C) }  x^{19}+2018x^{13}+1  \qquad  \textbf{(D) }  x^{17}+2018x^{13}+1 \qquad  \textbf{(E) }  2019x+2018 </math>
  
==Solution==
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==Solution 1==
  
 
We can see that our real solution has to lie in the open interval <math>(-1,0)</math>. From there, note that <math>x^a < x^b</math> if a, b are odd positive integers so <math>a<b</math>, so hence it can only either be B or E(as all of the other polynomials will be larger than the polynomial B). Finally, we can see that plugging in the root of <math>2019x+2018</math> into B gives a negative, and so the answer is <math>\fbox{B}</math>. (cpma213)
 
We can see that our real solution has to lie in the open interval <math>(-1,0)</math>. From there, note that <math>x^a < x^b</math> if a, b are odd positive integers so <math>a<b</math>, so hence it can only either be B or E(as all of the other polynomials will be larger than the polynomial B). Finally, we can see that plugging in the root of <math>2019x+2018</math> into B gives a negative, and so the answer is <math>\fbox{B}</math>. (cpma213)
  
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==Solution 2 (Calculus justification)==
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Note that <math>a(-1)=b(-1)=c(-1)=d(-1) < 0</math> and <math>a(0)=b(0)=c(0)=d(0) > 0</math>. Calculating the definite integral for each function on the interval <math>[-1,0]</math> we see that <math>B(x)\rvert^{0}_{-1}</math> 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 <math>\fbox{B}</math>.
  
 
==See Also==
 
==See Also==
 
{{AMC12 box|year=2018|ab=A|num-b=20|num-a=22}}
 
{{AMC12 box|year=2018|ab=A|num-b=20|num-a=22}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 01:11, 24 February 2018

Problem

Which of the following polynomials has the greatest real root? $\textbf{(A) }   x^{19}+2018x^{11}+1   \qquad        \textbf{(B) }   x^{17}+2018x^{11}+1   \qquad    \textbf{(C) }   x^{19}+2018x^{13}+1   \qquad   \textbf{(D) }  x^{17}+2018x^{13}+1 \qquad  \textbf{(E) }   2019x+2018$

Solution 1

We can see that our real solution has to lie in the open interval $(-1,0)$. From there, note that $x^a < x^b$ if a, b are odd positive integers so $a<b$, so hence it can only either be B or E(as all of the other polynomials will be larger than the polynomial B). Finally, we can see that plugging in the root of $2019x+2018$ into B gives a negative, and so the answer is $\fbox{B}$. (cpma213)

Solution 2 (Calculus justification)

Note that $a(-1)=b(-1)=c(-1)=d(-1) < 0$ and $a(0)=b(0)=c(0)=d(0) > 0$. Calculating the definite integral for each function on the interval $[-1,0]$ we see that $B(x)\rvert^{0}_{-1}$ 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 $\fbox{B}$.

See Also

2018 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 20
Followed by
Problem 22
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All AMC 12 Problems and Solutions

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