Two Functions Having Something Common

Algebra Level 4

Given that $f(x)=\sin x$ and $g(x)=x/10$ . If $x_{1},x_{2},...,x_{n}$ are the values of $x$ for which $f(x)=g(x)$ , then find the maximum possible value of $\dfrac {\left\lfloor x_{1}\right\rfloor- \left\lfloor x_{2}\right\rfloor+\left\lfloor x_{3}\right\rfloor-\left\lfloor x_{4}\right\rfloor+\cdots\pm \left\lfloor x_{n}\right\rfloor-1}{n-1}.$

Details and Assumptions

$\left\lfloor\cdot\right\rfloor$ represents the Floor Function .

The answer is 6.

1 solution

Chew-Seong Cheong
Oct 17, 2014

There are $7$ solutions for $\sin {x} = \dfrac {x}{10}$ .

They are: $x = 0, \pm 2.852, \pm 7.068, \pm 8.423$

$\Rightarrow \lfloor x \rfloor = -9, -8, -3, 0, 2, 7, 8$

The maximum possible value is:

$\dfrac {-(-9)-(-8)-(-3)+0+2+ 7+8-1} {7-1} = \dfrac {36}{6} = \boxed {6}$

How you got those 7 solutions except 0 ?

Karan Siwach - 6 years, 7 months ago

I used a spreadsheet and plotted $y = \sin {x}$ and $y = \frac {x}{10}$ . We only need an approximate solutions because we only need the greatest integer values.

Chew-Seong Cheong - 6 years, 7 months ago

by drawing graph of both sinx and X/10 and their comman points are solution

Akshay Sharma - 5 years, 7 months ago

Two Functions Having Something Common

The answer is 6.

1 solution

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