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

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Bunuel
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For all values, [x] denotes the least integer greater than or equal to [#permalink] New post   06 Jul 2020, 02:02
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For all values, \([x]\) denotes the least integer greater than or equal to x. If \(-2.5 < x< 1.5\), what is the least possible value of \([2x] + [x^2]\)?

A. -2
B. -1
C. 0
D. 1
E. 2
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B
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Re: For all values, [x] denotes the least integer greater than or equal to [#permalink] New post   06 Jul 2020, 02:24
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If −2.5<x<1.5, what is the least possible value of [2x]+[x^2]?

Let Value of [x] = -3,-2,-1,0 and 1
putting values one by one

Case 1: [x] = -3 in [2x]+[x^2]
=> [-6]+[9]
=> 3 (not the least value)

Case 2: [x] = -2 in [2x]+[x^2]
=> [-4]+[4]
=> 0 (not the least value)

Case 1: [x] = -1 in [2x]+[x^2]
=> [-2]+[1]
=> -1 (least value)

Case 1: [x] = 0 in [2x]+[x^2]
=> [0]+[0]
=> 0 (not the least value)

Case 1: [x] = 1 in [2x]+[x^2]
=> [2]+[1]
=> 3 (not the least value)

Hence answer is B
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Re: For all values, [x] denotes the least integer greater than or equal to [#permalink] New post   06 Jul 2020, 02:28
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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 [2x] + [x^2] = 0, -1, 0, 3
Therefore the least value is -1

Hence, OA is (B).
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Re: For all values, [x] denotes the least integer greater than or equal to [#permalink] New post   06 Jul 2020, 05:54
Kindly see the attachment.

CONCEPT: Greatest Integer and Range
IMO B
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Re: For all values, [x] denotes the least integer greater than or equal to [#permalink] New post   06 Jul 2020, 07:22
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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
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Re: For all values, [x] denotes the least integer greater than or equal to [#permalink] New post   06 Jul 2020, 09:20
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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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Re: For all values, [x] denotes the least integer greater than or equal to [#permalink] New post   06 Jul 2020, 15:10
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integer values that x can take = -2,-1,0,1
[2x]+[x^2]
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.
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Re: For all values, [x] denotes the least integer greater than or equal to [#permalink] New post   06 Jul 2020, 16:15
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For all values, [x][x] 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 [2x]+[x2][2x]+[x2]?

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.
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Re: For all values, [x] denotes the least integer greater than or equal to [#permalink] New post   06 Jul 2020, 22:32
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Ans - B

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

Check values of [2x]+[x2]
For [x]=-2 , value is 0
For [x]=-1, value is -1
For [x]=0, value is 0
For [x]=1, value is 3
For [x]=2, value is 8

So the least value is -1

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Re: For all values, [x] denotes the least integer greater than or equal to [#permalink] New post   21 Aug 2023, 07:42
Asked: For all values, \([x]\) denotes the least integer greater than or equal to x. If \(-2.5 < x< 1.5\), what is the least possible value of \([2x] + [x^2]\)?

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

IMO B
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Re: For all values, [x] denotes the least integer greater than or equal to [#permalink]
21 Aug 2023, 07:42
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