Find the greatest integer less than or equal to k

Question

It is given that  34 - |x^2 - 3x + 8| \leq k 
Find the greatest integer less than or equal to k

A. 27
B. 28
C. 29
D. 30
E. 31

Source : Indian B-School Entrance exam

CrackverbalGMAT · Accepted Answer

The value of k depends on the value of x.

The larger the value in the modulus, the smaller will be the difference between 34 and this value.

To get the maximum value of k, we need to get the least value for| x^2 −3x+8|

For negative values of x, the value in the modulus will always keep increasing. Negative values are ruled out.

When x = 0, then | x^2 −3x+8| = 8

When x = 1, | x^2 −3x+8| = 6

When x = 2, | x^2 −3x+8| = 6

When x = 3, | x^2 −3x+8| = 8

From x = 4 onwards, the value in the modulus keeps on increasing

The least value of | x^2 −3x+8| is 6.

Therefore the maximum value of k = 34 - 6 = 28

Option B

Arun Kumar

chetan2u · Answer

We have to find the minimum value of  x^2-3x+8 , and for this we can carry out differentiation => \frac{d}{d(x)}(x^2-3x+8)=0......2x-3=0.....x=\frac{3}{2}

The value of  x^2-3x+8=\frac{9}{4}-\frac{9}{2}+8=\frac{23}{4}=5.75

So  34-5.75\leq{k}....28.25\leq{k} 
Thus, integer value less than or equal to k is 28

B

BrushMyQuant · Answer

↧↧↧ Detailed Video Solution to the Problem ↧↧↧

https://www.youtube.com/watch?v=i62YKG0FbIU

Given that  34 - |x^2 - 3x + 8| \leq k  and we need to find the greatest integer less than or equal to k

Now on the left hand side we have 34 - | Of a Number | and we need to find the greatest possible value of this expression.
 | Of a Number | will be non-negative 
=> The output of | of a Number | will anyways get subtracted from 34 and will reduce the overall value.
So, the idea is to make the value inside the Absolute Value minimum

|x^2 - 3x + 8|  =  |x^2 - 2*\frac{3}{2}*x + (\frac{3}{2})^2 - (\frac{3}{2})^2 + 8| 
=  |(x - \frac{3}{2})^2 + 8 - \frac{9}{4}|  =  |(x - \frac{3}{2})^2 + \frac{23}{4}|

Now, this will be minimum when  (x - \frac{3}{2})^2  = 0 
=> x =  \frac{3}{2}

=>  34 - |x^2 - 3x + 8| \leq k  =  34 - |0 + \frac{23}{4}| \leq k  =  34 - |5.75| \leq k =  34 - 5.75 \leq k  =  28.25 \leq k 
=> k = 28

So,  Answer will be B 
Hope it helps!

Watch the following video to Master Absolute Values

https://www.youtube.com/watch?v=huFiBk8PaGY

KuhuPyasi · Answer

For the greatest value of k, |x2 - 3x + 8| should be minimum. The minimum value of the quadratic equation is given by (-D/4a when D= b2-4ac), therefore minimum value of quadratic eq. = 23/4.Substituting this value into the given equation yields the final answer of 28.

Adit_ · Answer

You can find greatest number we need to minimize the mod function inside:
One way to find min/max = differentiation the function = 
2x-3=0 
x=3/2

Now if you substitute x=3/2 you will get 5.75 as answer. You can see this is minimum as taking a simple x=0 gives value as 8 and x=1 gives 6.
Keeping 3/2 we see 5.75 is minimum value which means our integer value is 34-6 at the greatest.
34-6 =  28

Answer:  Option B

_________________________

It is not needed to do all this but these simple tricks can save lot of time. You may try out the parabola plotting and do -b/2a find vertex which will give max as parabola is opening upward. 
Whichever is suitable. IMO never settle for just what is required for GMAT because one can solve question like these using tricks that is outside portions!

Find the greatest integer less than or equal to k

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Find the greatest integer less than or equal to k

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