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Manager  Joined: 03 Jun 2010
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Location: United States (MI)
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If x^4 + y^4 = 100, then the greatest possible value of x is between  [#permalink]

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Question Stats: 57% (01:18) correct 43% (01:25) wrong based on 1159 sessions

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If x^4 + y^4 = 100, then the greatest possible value of x is between

A. 0 and 3
B. 3 and 6
C. 6 and 9
D. 9 and 12
E. 12 and 15

How (B) could be an answer?
Consider x=5.5, then x^4 is already bigger than 100. And y^4 can't be -ve.

Originally posted by ulm on 07 Aug 2010, 09:25.
Last edited by Bunuel on 22 Feb 2019, 07:17, edited 2 times in total.
Formatted the question
Math Expert V
Joined: 02 Sep 2009
Posts: 60727
Re: If x^4 + y^4 = 100, then the greatest possible value of x is between  [#permalink]

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16
ulm wrote:
How (B) could be an answer?
Consider x=5.5, then x^4 is already bigger than 100. And y^4 can't be -ve.

If x^4+y^4=100, then the greatest possible value of x is between
A. 0 and 3
B. 3 and 6
C. 6 and 9
D. 9 and 12
E. 12 and 15

General rule for such kind of problems:
to maximize one quantity, minimize the others;
to minimize one quantity, maximize the others.

So, to maximize $$x$$ we should minimize $$y^4$$. Least value of $$y^4$$ is zero. In this case $$x^4+0=100$$ --> $$x^4=100$$ --> $$x^2=10$$ --> $$x=\sqrt{10}\approx{3.2}$$, which is in the range (3,6).

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Veritas Prep GMAT Instructor V
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Re: If x^4 + y^4 = 100, then the greatest possible value of x is between  [#permalink]

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$$x^4 + y^4 = 100$$
When you see even powers, first thing that should come to your mind is that the term will be positive or zero.
If you want to maximize x in the sum, you should minimize y^4 so that this term's contribution in 100 is minimum possible. Since it is an even power, its smallest value is 0 when y = 0.

Then $$x^4$$ = 100
Since $$3^4 = 81$$ and $$4^4 = 256$$,x will lie between 3 and 4.
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Re: If x^4 + y^4 = 100, then the greatest possible value of x is between  [#permalink]

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Step 1:
x^4+y^4=100

x^4 will be maximum when y^4 is minimum. Lets assume y=0.1 so, y^4=0.0001

Step 2:

x^4+1^4=100
=> x^4+0.0001=100
=>x^4=100-0.0001
=>x^4=99.9999

Lets substitute x=3, i.e 3*3*3*3 = 81

So the value of x can be little more than 3 because 4*4*4*4=256

So the answer is option (B)
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Re: If x^4 + y^4 = 100, then the greatest possible value of x is between  [#permalink]

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Re: If x^4 + y^4 = 100, then the greatest possible value of x is between  [#permalink]

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Two things that we must consider in order to solve this problem are:

a) We do not look for an integer

b) We do not look for a specific number but we want to see the number we are looking for in what range falls....e.x it is positive ot it is greater than 10.....in our example all the answers give range....

solution has been given by minimizing Y meaning Y=0 Math Expert V
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Re: If x^4 + y^4 = 100, then the greatest possible value of x is between  [#permalink]

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Re: If x^4 + y^4 = 100, then the greatest possible value of x is between  [#permalink]

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boomtangboy wrote:
If x^4 + y^4 = 100, then the greatest possible value of x is between:

A. 0 and 3
B. 3 and 6
C. 6 and 9
D. 9 and 12
E. 12 and 15

For y = 0, $$x^4 = 100$$

$$3^4 = 81 & 4^4 = 256$$

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Re: If x^4 + y^4 = 100, then the greatest possible value of x is between  [#permalink]

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1
Hi All,

The answers to this question provide a great 'hint' as to how to go about solving it; since they're all essentially 'ranges', you can use them to figure out which solution contains the maximum value of X.

We're told that X^4 + Y^4 = 100. To maximize the value of X, we need to minimize the value of Y^4. The smallest that Y^4 could be is 0 (when Y = 0), so we'll have....

X^4 = 100

Looking at the answers, it makes sense to see what 3^4 equals....

3^4 = 81

Since that is BELOW 100, and 6^4 will clearly be MORE than 100, we have the correct answer.

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Re: If x^4 + y^4 = 100, then the greatest possible value of x is between  [#permalink]

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boomtangboy wrote:
If x^4 + y^4 = 100, then the greatest possible value of x is between:

A. 0 and 3
B. 3 and 6
C. 6 and 9
D. 9 and 12
E. 12 and 15

In an equation containing more than one variables, if we need to maximize one, it means that we have to minimize the others.
Since we have an even power of x and y, the minimum value of the variables can be 0

Therefore to maximize x, we need to minimize y.
Putting y = 0, x^4 = 100,
Or x = 3.xx

Hence Option C
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Re: If x^4 + y^4 = 100, then the greatest possible value of x is between  [#permalink]

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The key to solving this problem for me was understanding that x is the maximum, when y is the minimum. Since y is raised to the fourth power, the smallest y can be is 0.

Here is the full solution:
$$x^4+y^4=100$$
$$x^4+0=100$$
$$x^4=100$$
$$x^2=10$$
$$x=\sqrt{10}$$
$$4>x>3$$

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Re: If x^4 + y^4 = 100, then the greatest possible value of x is between  [#permalink]

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1
ulm wrote:
If x^4 + y^4 = 100, then the greatest possible value of x is between

A. 0 and 3
B. 3 and 6
C. 6 and 9
D. 9 and 12
E. 12 and 15

In determining the greatest possible value of x, we want to minimize y^4. Since the minimum value of y^4 is 0 (when y = 0), we have:

x^4 + 0 = 100

x^4 = 100

x^2 = 10

x = +/-√10

x ≈ 3.2 or -3.2

Thus, we see that the greatest value of x is between 3 and 6.

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GMAT 1: 690 Q50 V34 Re: If x^4 + y^4 = 100, then the greatest possible value of x is between  [#permalink]

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1
ulm wrote:
If x^4 + y^4 = 100, then the greatest possible value of x is between

A. 0 and 3
B. 3 and 6
C. 6 and 9
D. 9 and 12
E. 12 and 15

How (B) could be an answer?
Consider x=5.5, then x^4 is already bigger than 100. And y^4 can't be -ve.

Given: x^4 + y^4 = 100

Asked: The greatest possible value of x is between

For maximum x; y = 0
x^4 = 100
$$x = \sqrt{10} = 3.2$$

IMO B
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Re: If x^4 + y^4 = 100, then the greatest possible value of x is between  [#permalink]

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Whenever I see a^2 +b^2= c questions, I am reminded of th circle equation.

So in this case (x^2)^2 + (y^2)^2 =10^2

So since x^2 is a non negative value, I must lie between
0<=x^2<=10 .

So max(X) = √10 which lies between 3 and 4

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Joined: 20 Aug 2017
Posts: 104
If x^4 + y^4 = 100, then the greatest possible value of x is between  [#permalink]

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A common mistake in such type of questions can be that one assumes that x and y are integers.
so y cannot be 0. by assuming this the option becomes A.
but it is nowhere given that they are integers. and cannot be equal to 0.
so we maximise x. which gives us x = 100^1/4 and y = 0.
we get x = 3.16 If x^4 + y^4 = 100, then the greatest possible value of x is between   [#permalink] 24 Oct 2019, 06:01
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