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605-655 Level|   Algebra|   Min-Max Problems|            
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If x^4 + y^4 = 100, then the greatest possible value of x is between Updated on: 22 Feb 2019, 07:17
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.
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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

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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.
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B
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If x^4 + y^4 = 100, then the greatest possible value of x is between 07 Aug 2010, 09:40
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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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If x^4 + y^4 = 100, then the greatest possible value of x is between 26 Feb 2012, 10:45
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14. Min/Max Problems


General Discussion
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If x^4 + y^4 = 100, then the greatest possible value of x is between 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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If x^4 + y^4 = 100, then the greatest possible value of x is between 25 May 2011, 06:57
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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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If x^4 + y^4 = 100, then the greatest possible value of x is between 10 Apr 2012, 09:35
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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

:wink:
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If x^4 + y^4 = 100, then the greatest possible value of x is between 03 Jul 2013, 01:24
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Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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If x^4 + y^4 = 100, then the greatest possible value of x is between 22 Oct 2014, 02:56
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boomtangboy
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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If x^4 + y^4 = 100, then the greatest possible value of x is between 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.

Final Answer:
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B

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If x^4 + y^4 = 100, then the greatest possible value of x is between 01 Dec 2015, 02:41
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boomtangboy
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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If x^4 + y^4 = 100, then the greatest possible value of x is between 15 Feb 2017, 05:08
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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:
\(x^4+y^4=100\)
\(x^4+0=100\)
\(x^4=100\)
\(x^2=10\)
\(x=\sqrt{10}\)
\(4>x>3\)

The correct answer is B
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If x^4 + y^4 = 100, then the greatest possible value of x is between 16 Feb 2017, 12:14
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ulm
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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If x^4 + y^4 = 100, then the greatest possible value of x is between 15 Sep 2019, 11:14
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ulm
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

Show Spoiler
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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If x^4 + y^4 = 100, then the greatest possible value of x is between 17 Sep 2019, 06:55
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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

Posted from my mobile device
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If x^4 + y^4 = 100, then the greatest possible value of x is between 24 Oct 2019, 06:01
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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
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If x^4 + y^4 = 100, then the greatest possible value of x is between 25 Nov 2021, 06:06
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ulm
If x⁴ + y⁴ = 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 order to maximize the value of x, we must minimize the value of y⁴.
Since y⁴ ≥ 0 for all values of y, the minimum value of y⁴ occurs when y = 0.

If y = 0, we get: x⁴ + 0⁴ = 100
Simplify: : x⁴ = 100

3⁴ = 81 and 4⁴ = 256
Since 100 is between 81 and 256, we know that x is between 3 and 4

Answer: B
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If x^4 + y^4 = 100, then the greatest possible value of x is between 29 Jul 2023, 06:42
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For x to be the maximum in this case, y^4 has to be minimum therefore take y=0

Now X^4=100

since 3^4=81 and 4^4=256

Therefore 3<x<4

In options there is 3<x<6 which is correct

Hence B
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If x^4 + y^4 = 100, then the greatest possible value of x is between 16 Feb 2024, 02:53
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I did the following:
If x^4 + y^4 = 10^2, then x^2 + y^2 = 10
Possible values= 0, 1, 2, and 3. As 4^2=16 (>10)
Hence x has to be <4 but > than 3

Hope it it useful for someone :)
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If x^4 + y^4 = 100, then the greatest possible value of x is between 19 Feb 2025, 02:54
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