\(x^4 + y^4 < 10, 000\) and we know that x and y are positive integers and we need to find the greatest possible value of x

10,000 = \(10^4\) and to find the maximum value of x we need to keep value of y as minimum.
As y is a positive integer so minimum value of y can be 1
=> \(x^4 + 1^4 < 10, 000\)
=> \(x^4 + 1 < 10, 000\)
=> \(x^4 < 10, 000 - 1\)
=> \(x^4 < 9,999\)
We know that \(10^4\) = 10, 000
So, x will be just less than 10
=> Maximum value of x will be very close to 10 but less than 10
So, Maximum value of x will be 9 ( as x is an integer)

Answer can be D if the range is 8 to 12. (As between 9 and 12 doesn't mean that 9 and 12 are included)
Hope it helps!

To determine the largest possible value of x, you’ll need to determine the smallest possible value of y. If y is a positive integer, then y could be 1, making \(y^4 = 1\). Thus, x^4 could be a little less than 9,999, and the sum \(x^4 + y^4\) would still be less than 10,000. So, you need to find the approximate number that, raised to the fourth power, is equal to 9,999. That sounds like a tall order until you realize that you’re just approximating and that \(10, 000 = 10^4\)

So, 9,999 is approximately 10 raised to the fourth power, so 40 the greatest possible value of x is a little bit less than 10, or choice (D), between 9 and 12.