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

1. Translate the problem requirements: We have the constraint \(\mathrm{x}^4 + \mathrm{y}^4 = 100\), where both x and y can be any real numbers. We need to find the range containing the maximum possible value of x.
2. Identify the optimization condition: Since both \(\mathrm{x}^4\) and \(\mathrm{y}^4\) are non-negative (fourth powers are always \(\geq 0\)), \(\mathrm{x}^4\) is maximized when \(\mathrm{y}^4\) is minimized, which occurs when \(\mathrm{y} = 0\).
3. Calculate the maximum value: When \(\mathrm{y} = 0\), we get \(\mathrm{x}^4 = 100\), so the maximum value of x is the fourth root of 100.
4. Estimate the fourth root and match to ranges: Use the relationship that \(\sqrt[4]{100} = \sqrt{10} \approx 3.16\), then check which answer choice contains this value.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we have and what we're looking for in everyday terms.

We have two numbers, x and y, and we know that when we raise each to the fourth power and add them together, we get 100. Think of it like this: imagine you have two containers, and the total volume when you combine what's in both containers always equals 100 units.

Now, we want to find the biggest possible value that x can have. It's like asking: if we're splitting 100 units between two containers, what's the most we can put in the first container?

Mathematically, our constraint is: \(\mathrm{x}^4 + \mathrm{y}^4 = 100\), where x and y can be any real numbers (positive, negative, or zero).

Our goal is to find the maximum possible value of x and determine which range it falls into.

Process Skill: TRANSLATE - Converting the mathematical constraint into an understandable optimization problem

2. Identify the optimization condition

Here's the key insight: since we're dealing with fourth powers, both \(\mathrm{x}^4\) and \(\mathrm{y}^4\) are always non-negative (zero or positive). This is because any real number raised to an even power gives a non-negative result.

Think about it this way: if you have a fixed total (100) that must be split between two non-negative quantities (\(\mathrm{x}^4\) and \(\mathrm{y}^4\)), then to make one as large as possible, you need to make the other as small as possible.

Since \(\mathrm{y}^4 \geq 0\), the smallest value \(\mathrm{y}^4\) can take is 0, which happens when \(\mathrm{y} = 0\).

Therefore, x is maximized when \(\mathrm{y} = 0\).

Process Skill: INFER - Recognizing that optimization occurs at the boundary condition where one variable equals zero

3. Calculate the maximum value

Now let's find the actual maximum value of x.

When \(\mathrm{y} = 0\), our equation becomes:
\(\mathrm{x}^4 + 0^4 = 100\)
\(\mathrm{x}^4 = 100\)

To find x, we need the fourth root of 100:
\(\mathrm{x} = \sqrt[4]{100}\)

We can rewrite this as:
\(\mathrm{x} = (100)^{1/4}\)

Since \(100 = 10^2\), we have:
\(\mathrm{x} = (10^2)^{1/4} = 10^{2/4} = 10^{1/2} = \sqrt{10}\)

So the maximum value of x is \(\sqrt{10}\).

4. Estimate the fourth root and match to ranges

Now we need to estimate \(\sqrt{10}\) to see which answer choice it falls into.

We know that:

• \(\sqrt{9} = 3\)
• \(\sqrt{16} = 4\)

Since 10 is between 9 and 16, \(\sqrt{10}\) is between 3 and 4.

For a more precise estimate:

• \(3^2 = 9\)
• \(3.2^2 = 10.24\)

So \(\sqrt{10}\) is approximately 3.16.

Looking at our answer choices:

• A: 0 and 3 (doesn't include 3.16)
• B: 3 and 6 (includes 3.16) ✓
• C: 6 and 9 (too high)
• D: 9 and 12 (too high)
• E: 12 and 15 (too high)

4. Final Answer

The greatest possible value of x is \(\sqrt{10} \approx 3.16\), which falls in the range between 3 and 6.

Therefore, the answer is B: 3 and 6.

Common Faltering Points

Errors while devising the approach

Students often assume that x and y must be positive numbers only, not realizing that they can be any real numbers (positive, negative, or zero). This misconception can lead them to think the problem is more complex than it actually is, potentially missing the key insight about optimization.

Some students may try to solve for specific values of x and y that satisfy the equation, rather than understanding that we need to find the maximum possible value of x. They might attempt to find all solutions instead of focusing on the optimization aspect.

Students may not immediately recognize that since both \(\mathrm{x}^4\) and \(\mathrm{y}^4\) are always non-negative (due to even powers), the sum has specific properties that make optimization straightforward. This insight is crucial for determining when x reaches its maximum value.

Errors while executing the approach

When solving \(\mathrm{x}^4 = 100\), students might confuse this with finding the square root instead of the fourth root. They might incorrectly calculate \(\mathrm{x} = \sqrt{100} = 10\) instead of \(\mathrm{x} = \sqrt[4]{100} = \sqrt{10}\).

Students may make mistakes when simplifying \((100)^{1/4}\). They might incorrectly handle the exponent rules, such as writing \((10^2)^{1/4} = 10^{2×4}\) instead of \(10^{2÷4} = 10^{1/2}\).

When estimating \(\sqrt{10}\), students might use incorrect reference points or make arithmetic mistakes. For example, they might incorrectly estimate that \(\sqrt{10}\) is closer to 2.5 or 4.5 instead of recognizing it's approximately 3.16.

Errors while selecting the answer

Students might be unsure whether \(\sqrt{10} \approx 3.16\) belongs in interval "0 and 3" or "3 and 6" since 3.16 is very close to 3. They need to understand that the interval "3 and 6" typically means \(3 < \mathrm{x} < 6\), which includes 3.16.

If students roughly estimate \(\sqrt{10}\) as "a little more than 3" but don't calculate precisely enough, they might incorrectly choose option A (0 and 3) thinking that "a little more than 3" still falls in that range.