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

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

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New post Updated on: 10 Jan 2016, 16:01
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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 ENGRTOMBA2018 on 10 Jan 2016, 16:01, edited 1 time in total.
Formatted the question
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Re: If x^4 + y^4 = 100, then the greatest possible value of x  [#permalink]

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New post 07 Aug 2010, 09:40
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ulm wrote:
Please find the attached pict.
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).

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

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New post 28 Oct 2010, 19:35
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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  [#permalink]

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New post 25 May 2011, 06:57
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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If x^4 + y^4 = 100, then the greatest possible value of x  [#permalink]

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

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New post 10 Apr 2012, 09:35
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

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

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New post 03 Jul 2013, 01:24
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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

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New post 22 Oct 2014, 02:56
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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, \(x^4 = 100\)

\(3^4 = 81 & 4^4 = 256\)

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

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New post 30 Nov 2015, 12:07
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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  [#permalink]

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New post 01 Dec 2015, 02:41
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  [#permalink]

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New post 15 Feb 2017, 05:08
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:
\(x^4+y^4=100\)
\(x^4+0=100\)
\(x^4=100\)
\(x^2=10\)
\(x=\sqrt{10}\)
\(4>x>3\)

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

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New post 16 Feb 2017, 12:14
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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.

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

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New post 03 Aug 2018, 04:26
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Re: If x^4 + y^4 = 100, then the greatest possible value of x &nbs [#permalink] 03 Aug 2018, 04:26
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