Difference between revisions of "MIE 2016/Day 1/Problem 8"

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==Solution ==
 
==Solution ==
 
The minimum value of this function is when <math>x</math> is in the middle between 1 and 2017. <math>\frac{2017+1}{2}=1009</math> so inputting <math>1009</math> as <math>x</math>:
 
The minimum value of this function is when <math>x</math> is in the middle between 1 and 2017. <math>\frac{2017+1}{2}=1009</math> so inputting <math>1009</math> as <math>x</math>:
<cmath>f(1009)=sqrt{1008+1007+\dots+1+0+1+\dots+1007+1008}=sqrt{1008*1009}\approx 1008</cmath>
+
<cmath>f(1009)=\sqrt{1008+1007+\dots+1+0+1+\dots+1007+1008}=\sqrt{1008\times 1009}\approx 1008</cmath>
  
 
==See Also==
 
==See Also==

Revision as of 13:00, 9 May 2024

Problem 8

Let $f(x)=\sqrt{|x-1|+|x-2|+|x-3|+...+|x-2017|}$. The minimum value of $f(x)$ is in the interval:

(a) $(-\infty,1008]$

(b) $(1008,1009]$

(c) $(1009,1010]$

(d) $(1010,1011]$

(e) $(1011,+\infty)$

Solution

The minimum value of this function is when $x$ is in the middle between 1 and 2017. $\frac{2017+1}{2}=1009$ so inputting $1009$ as $x$: \[f(1009)=\sqrt{1008+1007+\dots+1+0+1+\dots+1007+1008}=\sqrt{1008\times 1009}\approx 1008\]

See Also