GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 16 Oct 2018, 05:13

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# What is the maximum value of the expression 22/(4x^4 - 32x^2 + 75)

Author Message
TAGS:

### Hide Tags

Intern
Joined: 23 Mar 2014
Posts: 16
What is the maximum value of the expression 22/(4x^4 - 32x^2 + 75)  [#permalink]

### Show Tags

30 Dec 2016, 10:47
5
00:00

Difficulty:

45% (medium)

Question Stats:

70% (02:50) correct 30% (02:29) wrong based on 77 sessions

### HideShow timer Statistics

What is the maximum value of the expression $$\frac{22}{4x^4 - 32x^2 + 75}$$

A) 1
B) 10/9
C) 2
D) 23/11
E) 25/11
Senior Manager
Joined: 13 Oct 2016
Posts: 367
GPA: 3.98
Re: What is the maximum value of the expression 22/(4x^4 - 32x^2 + 75)  [#permalink]

### Show Tags

30 Dec 2016, 11:33
Rocky1304 wrote:
what is the maximum value of the expression
$$22/(4x^4-32x^2+75)$$

A) 1
B) 10/9
C)2
D) 23/11
E) 25/11

$$\frac{22}{(4x^4-32x^2+75)}$$

This expression will take its maximum value when $$4x^4-32x^2+75$$ will take its minimum value.

$$4x^4-32x^2+75 = (2x^2)^2 + 2*2x^2*8 + 64 + 11 = (2x^2 - 8)^2 + 11$$

If we get negative or positive value of $$2x^2 - 8$$ it will be squared and become positive, hence increasing our denominator and decreasing value of our fraction. The only possible minimum can be achieved when $$2x^2 - 8 = 0$$

$$2x^2 - 8 =0$$

$$x^2 = 4$$

$$x= +/- 2$$

e-GMAT Representative
Joined: 04 Jan 2015
Posts: 2063
Re: What is the maximum value of the expression 22/(4x^4 - 32x^2 + 75)  [#permalink]

### Show Tags

31 Dec 2016, 00:02
vitaliyGMAT wrote:
Rocky1304 wrote:
what is the maximum value of the expression
$$22/(4x^4-32x^2+75)$$

A) 1
B) 10/9
C)2
D) 23/11
E) 25/11

$$\frac{22}{(4x^4-32x^2+75)}$$

This expression will take its maximum value when $$4x^4-32x^2+75$$ will take its minimum value.

$$4x^4-32x^2+75 = (2x^2)^2 + 2*2x^2*8 + 64 + 11 = (2x^2 - 8)^2 + 11$$

If we get negative or positive value of $$2x^2 - 8$$ it will be squared and become positive, hence increasing our denominator and decreasing value of our fraction. The only possible minimum can be achieved when $$2x^2 - 8 = 0$$

$$2x^2 - 8 =0$$

$$x^2 = 4$$

$$x= +/- 2$$

Hey,

I don't think there was any need to find the value of x in this case. As correctly pointed by you the value of the expression will be maximum when the denominator is minimum and to do that we need to need ensure that $$2x^2 - 8 = 0$$

Thus the maximum value of the expression is :

$$22/(4x^4-32x^2+75)$$
$$={22}{(2x^2 - 8)^2 + 11}$$
$$=22/11$$
$$= 2$$

I think it just a coincidence that the value of x and the value of the expression is 2. And maybe that is why you did not notice that my mistake you found the value of x instead of the value of the expression. Just thought to let you know.

Thanks,
Saquib
Quant Expert
e-GMAT

To practise ten 700+ Level Number Properties Questions attempt the The E-GMAT Number Properties Knockout

_________________

Number Properties | Algebra |Quant Workshop

Success Stories
Guillermo's Success Story | Carrie's Success Story

Ace GMAT quant
Articles and Question to reach Q51 | Question of the week

Number Properties – Even Odd | LCM GCD | Statistics-1 | Statistics-2
Word Problems – Percentage 1 | Percentage 2 | Time and Work 1 | Time and Work 2 | Time, Speed and Distance 1 | Time, Speed and Distance 2
Advanced Topics- Permutation and Combination 1 | Permutation and Combination 2 | Permutation and Combination 3 | Probability
Geometry- Triangles 1 | Triangles 2 | Triangles 3 | Common Mistakes in Geometry
Algebra- Wavy line | Inequalities

Practice Questions
Number Properties 1 | Number Properties 2 | Algebra 1 | Geometry | Prime Numbers | Absolute value equations | Sets

| '4 out of Top 5' Instructors on gmatclub | 70 point improvement guarantee | www.e-gmat.com

Senior Manager
Joined: 13 Oct 2016
Posts: 367
GPA: 3.98
Re: What is the maximum value of the expression 22/(4x^4 - 32x^2 + 75)  [#permalink]

### Show Tags

31 Dec 2016, 01:39
EgmatQuantExpert wrote:
vitaliyGMAT wrote:
Rocky1304 wrote:
what is the maximum value of the expression
$$22/(4x^4-32x^2+75)$$

A) 1
B) 10/9
C)2
D) 23/11
E) 25/11

$$\frac{22}{(4x^4-32x^2+75)}$$

This expression will take its maximum value when $$4x^4-32x^2+75$$ will take its minimum value.

$$4x^4-32x^2+75 = (2x^2)^2 + 2*2x^2*8 + 64 + 11 = (2x^2 - 8)^2 + 11$$

If we get negative or positive value of $$2x^2 - 8$$ it will be squared and become positive, hence increasing our denominator and decreasing value of our fraction. The only possible minimum can be achieved when $$2x^2 - 8 = 0$$

$$2x^2 - 8 =0$$

$$x^2 = 4$$

$$x= +/- 2$$

Hey,

I don't think there was any need to find the value of x in this case. As correctly pointed by you the value of the expression will be maximum when the denominator is minimum and to do that we need to need ensure that $$2x^2 - 8 = 0$$

Thus the maximum value of the expression is :

$$22/(4x^4-32x^2+75)$$
$$={22}{(2x^2 - 8)^2 + 11}$$
$$=22/11$$
$$= 2$$

I think it just a coincidence that the value of x and the value of the expression is 2. And maybe that is why you did not notice that my mistake you found the value of x instead of the value of the expression. Just thought to let you know.

Thanks,
Saquib
Quant Expert
e-GMAT

To practise ten 700+ Level Number Properties Questions attempt the The E-GMAT Number Properties Knockout

Hi Saquib
Thanks for your kind advice. It was way beyond midnight and I did not put the solution to its logical end. What I’ve found in fact is the value of x that will maximize our fraction. In general the proper way to solve questions like that, as it’s done in analysis, is to find first and second derivatives of denominator and extreme values of our variable, put them into fraction to find its max or min value. But, calculus is beyond gmat, that’s why we have this case where we can simplify our expression to “perfect square plus 11” and, as you’ve correctly mentioned, it’s quite enough to take our perfect square equal to zero. I just had to put that x into our fraction, but that was unnecessary, 22/11 will be our answer. In general, this won’t do and we’ll need to go the long way.
Thanks again!
Merry holidays
Vitaliy
Intern
Joined: 17 Jun 2018
Posts: 11
Location: India
Schools: IMD '20
GPA: 2.84
WE: Engineering (Consulting)
Re: What is the maximum value of the expression 22/(4x^4 - 32x^2 + 75)  [#permalink]

### Show Tags

19 Sep 2018, 12:14
In order to maximize the expression the denominator should be minimum.

For a quadratic equation to be minimum the d/dx[(4x^4-32x^2+75)]=16x^3 - 64x=0
Which gives x=+2,-2
Substitute this in the original expression and we get answer as (c).
Intern
Joined: 27 Oct 2017
Posts: 4
Location: United States
GPA: 3.59
Re: What is the maximum value of the expression 22/(4x^4 - 32x^2 + 75)  [#permalink]

### Show Tags

19 Sep 2018, 17:31
1
Can someone please break down how

$$4x^4$$−$$32x^2$$+75 becomes ($$2x^2$$−8)$$^{2}$$+11 ?

Thanks!
Manager
Joined: 14 Jun 2018
Posts: 100
Re: What is the maximum value of the expression 22/(4x^4 - 32x^2 + 75)  [#permalink]

### Show Tags

20 Sep 2018, 03:14
2
RichardSaunders wrote:
Can someone please break down how

$$4x^4$$−$$32x^2$$+75 becomes ($$2x^2$$−8)$$^{2}$$+11 ?

Thanks!

you need to think about $$(a-b)^2 = a^2 - 2ab + b^2$$
$$4x^4−32x^2+75$$

$$4x^4$$ can be weritten as $$( 2 x^2 ) ^ 2$$ = $$a^2$$

$$- 32x^2$$ = 2 * $$2x^2$$ * (-8) = 2ab (you need 32/4 = 8)

$$b^2 = -8^2 = 64$$ (you can separate 75 into 64 + 11)

Therefore $$(2x^2 - 8)^ + 11$$
Intern
Joined: 27 Oct 2017
Posts: 4
Location: United States
GPA: 3.59
Re: What is the maximum value of the expression 22/(4x^4 - 32x^2 + 75)  [#permalink]

### Show Tags

20 Sep 2018, 05:33
Thanks pandeyashwin, very simple/clear
Re: What is the maximum value of the expression 22/(4x^4 - 32x^2 + 75) &nbs [#permalink] 20 Sep 2018, 05:33
Display posts from previous: Sort by