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Difference between revisions of "2019 AMC 10A Problems/Problem 19"

(→‎Solution 6 (no words))
(→‎Solution 3 (calculus))
 
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Letting <math>y=x^2+5x</math>, we get the expression <math>(y+4)(y+6)+2019</math>. Now, we can find the critical points of <math>(y+4)(y+6)</math> to minimize the function:
 
Letting <math>y=x^2+5x</math>, we get the expression <math>(y+4)(y+6)+2019</math>. Now, we can find the critical points of <math>(y+4)(y+6)</math> to minimize the function:
  
−
<math>\frac{d}{dx}(y^2+10y+24)=0</math>
+
<math>\frac{d}{dy}(y^2+10y+24)=0</math>
  
 
<math>2y+10=0</math>
 
<math>2y+10=0</math>
  
−
<math>2y(y+5)=0</math>
+
<math>2y=-5</math>
  
 
<math>y=-5,0</math>
 
<math>y=-5,0</math>
Line 35: Line 35:
  
 
''Note'': We could also have used the result that minimum/maximum point of a parabola <math>y = ax^2 + bx + c</math> occurs at <math>x=-\frac{b}{2a}</math>.
 
''Note'': We could also have used the result that minimum/maximum point of a parabola <math>y = ax^2 + bx + c</math> occurs at <math>x=-\frac{b}{2a}</math>.
 +
 +
''Note 2:'' This solution is somewhat "lucky", since when we define variables to equal a function, and create another function out of these variables, the domain of such function may vary from the initial one. This is important because the maximum and minimum value of a function is dependent on its domain, e.g:
 +
 +
<math>f(x)=x^2</math> has no maximum value in the the integers, but once restricting the domain to <math>(-5, 5)</math> the maximum value of <math>f(x)</math> is <math>25</math>.
 +
 +
Also, observe that if we were to evaluate this by taking the derivative of <math>(x+1)(x+2)(x+3)(x+4)+2019</math>, we would get <math>-5</math> as the <math>x</math>-value to obtain the minimum <math>y</math>-value of this expression. It can be seen that <math>-5</math> is actually an inflection point, instead of a minimum or maximum.
 +
 +
-Note 2 from Benedict T (countmath1)
  
 
==Solution 4(guess with answer choices)==
 
==Solution 4(guess with answer choices)==
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Answer choices <math>C</math>, <math>D</math>, and <math>E</math> are impossible, since <math>(x+1)(x+2)(x+3)(x+4)</math> can be negative (as seen when e.g. <math>x = -\frac{3}{2}</math>). Plug in <math>x = -\frac{3}{2}</math> to see that it becomes <math>2019 - \frac{15}{16}</math>, so round this to <math>\boxed{\textbf{(B) }2018}</math>.
 
Answer choices <math>C</math>, <math>D</math>, and <math>E</math> are impossible, since <math>(x+1)(x+2)(x+3)(x+4)</math> can be negative (as seen when e.g. <math>x = -\frac{3}{2}</math>). Plug in <math>x = -\frac{3}{2}</math> to see that it becomes <math>2019 - \frac{15}{16}</math>, so round this to <math>\boxed{\textbf{(B) }2018}</math>.
  
−
We can also see that the limit of the function is at least -1 since at the minimum, two of the numbers are less than 1, but two are between 1 and 2.
+
We can also see that the limit of the function is at least <math>-1</math> since at the minimum, two of the numbers are less than <math>1</math>, but two are between <math>1</math> and <math>2</math>.
  
 +
==Solution 6 (also calculus but more convoluted)==
 +
We can ignore the <math>2019</math> and consider it later, as it is a constant. By difference of squares, we can group this into <math>\left((x+2.5)^2-0.5^2\right)\left((x+2.5)^2-1.5^2\right)</math>. We pull a factor of <math>4</math> into each term to avoid dealing with decimals:
  
−
==Solution 6 (no words)==
+
<cmath>\dfrac{\left((2x+5)^2-1\right)\left((2x+5)^2-9\right)}{16}.</cmath>
−
<math>\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)+2019</math>
 
−
<math>=\left(x+2\right)\left(x+3\right)\left(x+1\right)\left(x+4\right)+2019</math>
 
−
<math>=\left(x^{2}+5x+6\right)\left(x^{2}+5x+4\right)+2019</math>
 
−
<math>=\left(\left(x^{2}+5x+5\right)+1\right)\left(\left(x^{2}+5x+5\right)-1\right)+2019</math>
 
−
<math>=\left(x^{2}+5x+5\right)^{2}-1^{2}+2019</math>.
 
  
−
<math>x^{2}+5x+5=0 \Rightarrow x=\frac{-5\pm\sqrt{25-20}}{2}=\frac{-5\pm\sqrt{5}}{2}</math>.
+
Now, we let <math>a=2x+5</math>. Our expression becomes:
  
−
<math>\left(x^{2}+5x+5\right)^{2}=0</math>
+
<cmath>\dfrac{(a-1)(a-9)}{16}=\dfrac{a^2-10a+9}{16}.</cmath>
  
−
<math>0-1^{2}+2019=2018 \Rightarrow \boxed{B}</math>.
+
Taking the derivative, we get <math>\dfrac{2a-10}{16}=\dfrac{a-5}8.</math> This is equal to <math>0</math> when <math>a=5</math>, and plugging in <math>a=5</math>, we get the expression is equal to <math>-1</math> and therefore our answer is <math>2019-1=\boxed{\text{(B)}~2018}.</math>
  
−
==Solution 7==
+
~Technodoggo
  
−
Arcticturn
 
  
 
==Video Solutions==
 
==Video Solutions==
 +
https://www.youtube.com/watch?v=Vf2LkM7ExhY by SpreadTheMathLove
 +
 
https://www.youtube.com/watch?v=Lis8yKT9WXc (less than 2 minutes)
 
https://www.youtube.com/watch?v=Lis8yKT9WXc (less than 2 minutes)
 
*https://youtu.be/NRa3VnjNVbw - Education, the Study of Everything
 
*https://youtu.be/NRa3VnjNVbw - Education, the Study of Everything

Latest revision as of 02:21, 2 September 2024

Problem

What is the least possible value of \[(x+1)(x+2)(x+3)(x+4)+2019\]where $x$ is a real number?

$\textbf{(A) } 2017 \qquad\textbf{(B) } 2018 \qquad\textbf{(C) } 2019 \qquad\textbf{(D) } 2020 \qquad\textbf{(E) } 2021$

Solution 1

Grouping the first and last terms and two middle terms gives $(x^2+5x+4)(x^2+5x+6)+2019$, which can be simplified to $(x^2+5x+5)^2-1+2019$. Noting that squares are nonnegative, and verifying that $x^2+5x+5=0$ for some real $x$, the answer is $\boxed{\textbf{(B) } 2018}$.

Solution 2

Let $a=x+\tfrac{5}{2}$. Then the expression $(x+1)(x+2)(x+3)(x+4)$ becomes $\left(a-\tfrac{3}{2}\right)\left(a-\tfrac{1}{2}\right)\left(a+\tfrac{1}{2}\right)\left(a+\tfrac{3}{2}\right)$.

We can now use the difference of two squares to get $\left(a^2-\tfrac{9}{4}\right)\left(a^2-\tfrac{1}{4}\right)$, and expand this to get $a^4-\tfrac{5}{2}a^2+\tfrac{9}{16}$.

Refactor this by completing the square to get $\left(a^2-\tfrac{5}{4}\right)^2-1$, which has a minimum value of $-1$. The answer is thus $2019-1=\boxed{\textbf{(B) }2018}$.

Solution 3 (calculus)

Similar to Solution 1, grouping the first and last terms and the middle terms, we get $(x^2+5x+4)(x^2+5x+6)+2019$.

Letting $y=x^2+5x$, we get the expression $(y+4)(y+6)+2019$. Now, we can find the critical points of $(y+4)(y+6)$ to minimize the function:

$\frac{d}{dy}(y^2+10y+24)=0$

$2y+10=0$

$2y=-5$

$y=-5,0$

To minimize the result, we use $y=-5$. Hence, the minimum is $(-5+4)(-5+6)=-1$, so $-1+2019 = \boxed{\textbf{(B) }2018}$.

Note: We could also have used the result that minimum/maximum point of a parabola $y = ax^2 + bx + c$ occurs at $x=-\frac{b}{2a}$.

Note 2: This solution is somewhat "lucky", since when we define variables to equal a function, and create another function out of these variables, the domain of such function may vary from the initial one. This is important because the maximum and minimum value of a function is dependent on its domain, e.g:

$f(x)=x^2$ has no maximum value in the the integers, but once restricting the domain to $(-5, 5)$ the maximum value of $f(x)$ is $25$.

Also, observe that if we were to evaluate this by taking the derivative of $(x+1)(x+2)(x+3)(x+4)+2019$, we would get $-5$ as the $x$-value to obtain the minimum $y$-value of this expression. It can be seen that $-5$ is actually an inflection point, instead of a minimum or maximum.

-Note 2 from Benedict T (countmath1)

Solution 4(guess with answer choices)

The expression is negative when an odd number of the factors are negative. This happens when $-2 < x < -1$ or $-4 < x < -3$. Plugging in $x = -\frac32$ or $x = -\frac72$ yields $-\frac{15}{16}$, which is very close to $-1$. Thus the answer is $-1 + 2019 = \boxed{\textbf{(B) }2018}$.

Solution 5 (using the answer choices)

Answer choices $C$, $D$, and $E$ are impossible, since $(x+1)(x+2)(x+3)(x+4)$ can be negative (as seen when e.g. $x = -\frac{3}{2}$). Plug in $x = -\frac{3}{2}$ to see that it becomes $2019 - \frac{15}{16}$, so round this to $\boxed{\textbf{(B) }2018}$.

We can also see that the limit of the function is at least $-1$ since at the minimum, two of the numbers are less than $1$, but two are between $1$ and $2$.

Solution 6 (also calculus but more convoluted)

We can ignore the $2019$ and consider it later, as it is a constant. By difference of squares, we can group this into $\left((x+2.5)^2-0.5^2\right)\left((x+2.5)^2-1.5^2\right)$. We pull a factor of $4$ into each term to avoid dealing with decimals:

\[\dfrac{\left((2x+5)^2-1\right)\left((2x+5)^2-9\right)}{16}.\]

Now, we let $a=2x+5$. Our expression becomes:

\[\dfrac{(a-1)(a-9)}{16}=\dfrac{a^2-10a+9}{16}.\]

Taking the derivative, we get $\dfrac{2a-10}{16}=\dfrac{a-5}8.$ This is equal to $0$ when $a=5$, and plugging in $a=5$, we get the expression is equal to $-1$ and therefore our answer is $2019-1=\boxed{\text{(B)}~2018}.$

~Technodoggo


Video Solutions

https://www.youtube.com/watch?v=Vf2LkM7ExhY by SpreadTheMathLove

https://www.youtube.com/watch?v=Lis8yKT9WXc (less than 2 minutes)

See Also

2019 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 18 		Followed by
Problem 20
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 10 Problems and Solutions

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