For all values, [x] denotes the least integer greater than or equal to

Question

Competition Mode Question

For all values,     denotes the least integer greater than or equal to x. If  -2.5 < x< 1.5 , what is the least possible value of    +   ?

A. -2 
B. -1
C. 0
D. 1
E. 2

v12345 · Accepted Answer

Values of x can be = -2, -1, 0, 1
Accordingly the values of 2x = -4, -2, 0 2
Accordingly the values of x^2 = 4, 1, 0, 1
Accordingly the value of   +   = 0, -1, 0, 3
Therefore the least value is -1

Hence, OA is (B).

yashikaaggarwal · Answer

If −2.5 + => 3 (not the least value) Case 2: = -2 in + => + => 0 (not the least value) Case 1: = -1 in + => + => -1 (least value) Case 1: = 0 in + => + => 0 (not the least value) Case 1: = 1 in + => + => 3 (not the least value) Hence answer is B

sambitspm · Answer

Kindly see the attachment.

CONCEPT: Greatest Integer and Range 
IMO B

MayankSingh · Answer

IMO B

−2.5 < x < 1.5

-5 < 2x < 3 & 0 < x^2 < 6.25

For any value of x>=0 , Function = [2x]+[x^2] >=0

So min value will be for x<0

At x=-2 & x=0 , f(x)=0 & f(-1)=-1
So function has a minima between -2 & 0.
Take any point in this range say , -1.6 , -1.4 , F(x)= [-3.2] + [2.56] = 0
but f(-1)= -1

So min value will be at -1 and the min value is also -1.

B. -1

alejogh89 · Answer

If x=-1, then:
2* (-1) + (-1)^2 = -1

All other values are greater than this.

So it would be B.

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AnubhaVerma · Answer

integer values that x can take = -2,-1,0,1
 + 
for x=-2 is 0
for x=-1 is -1
for x=0 is 0
for x=1 is 3
thus, the minimum value of the equation is -1.

minustark · Answer

For all values,    denotes the least integer greater than or equal to x. If −2.5<x<1.5−2.5<x<1.5, what is the least possible value of  +  + ?

A. -2
B. -1
C. 0
D. 1
E. 2

The least value x can take is -2.51, or -2. and the greatest value x can take is 1.49 or, 2. So x can assume -2, -1, 0 , 1, 2.
When x = -2, 2x + x^2 = -4 + 4 = 0, 
when x = -1, -2 + 1 = -1, 
when x = 0, 0
when x =1, the function is 3
when x =2, it becomes 6.

So, 2x + x^2 can take the least value of -1.
B is the answer.

Abi1995 · Answer

Ans - B

Given -2.5<x<1.5, so possible values of   as integers in this range are -2,-1,0,1,2

Check values of  + 
For  =-2 , value is 0
For  =-1, value is -1
For  =0, value is 0
For  =1, value is 3
For  =2, value is 8

So the least value is -1

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Kinshook · Answer

Asked: For all values,     denotes the least integer greater than or equal to x. If  -2.5 < x< 1.5 , what is the least possible value of    +   ?

+    = ( +1)^2 - 1
  = -1 is for -2 < x < -1
Least possible value of    +    = - 1

IMO B

Adit_ · Answer

This can be solved by simple differentiation in order to find min.

2x+2=0
x=-1
Now to check if a value of less than -1 then the positive value will be far more than negative as greatest value of a negative integer will pull it higher and greatest value of positive will push it to next number thereby increasing the value for any other value apart from x=-1.

Thus when x=-1 then -2+1 = -1.

Answer:  Option B

For all values, [x] denotes the least integer greater than or equal to

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For all values, [x] denotes the least integer greater than or equal to

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

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My Rewards