If x, y, z are real numbers such that `x+y+z=4` and `x^(2)+y^(2)+z^(2)=6,` then `x,y,z` lie in :

Options

A. `[(3)/(2),2]`
B. `[(2)/(3),2]`
C. `[0, (2)/(3)]`
D. none of these

Answer

To solve the problem, we need to analyze the given conditions step by step.

### Given:
1. \( x + y + z = 4 \) (Equation 1)
2. \( x^2 + y^2 + z^2 = 6 \) (Equation 2)

### Step 1: Use the identity for squares
We can use the identity that relates the sum of squares to the square of the sum:
\[ (x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + xz + yz) \]
Substituting the values from Equation 1 and Equation 2 into this identity:
\[ 4^2 = 6 + 2(xy + xz + yz) \]
This simplifies to:
\[ 16 = 6 + 2(xy + xz + yz) \]
\[ 16 - 6 = 2(xy + xz + yz) \]
\[ 10 = 2(xy + xz + yz) \]
\[ xy + xz + yz = 5 \quad \text{(Equation 3)} \]

### Step 2: Analyze the equations
Now we have three equations:
1. \( x + y + z = 4 \)
2. \( xy + xz + yz = 5 \)
3. \( x^2 + y^2 + z^2 = 6 \)

### Step 3: Consider possible values for \(x, y, z\)
We can assume \(x, y, z\) are the roots of the cubic polynomial \(t^3 - (x+y+z)t^2 + (xy+xz+yz)t - xyz = 0\). This gives us:
\[ t^3 - 4t^2 + 5t - xyz = 0 \]
We need to find the possible values for \(x, y, z\) that satisfy these equations.

### Step 4: Check for specific values
Let’s try some values that satisfy the first equation \(x + y + z = 4\):
- If we take \(x = 1\), \(y = 1\), and \(z = 2\):
- Check \(x + y + z = 1 + 1 + 2 = 4\) (True)
- Check \(x^2 + y^2 + z^2 = 1^2 + 1^2 + 2^2 = 1 + 1 + 4 = 6\) (True)

Thus, \(x = 1\), \(y = 1\), \(z = 2\) is one solution.

### Step 5: Determine the range of values
Now we need to determine the range of values for \(x, y, z\). Since \(x, y, z\) are real numbers and we have found a valid solution, we can analyze the options given in the question.

### Step 6: Evaluate the options
1. **Option 1:** \( \left[\frac{3}{2}, 2\right] \) - This range does not include 1.
2. **Option 2:** \( \left[\frac{2}{3}, 2\right] \) - This range includes values like 1 and 2.
3. **Option 3:** \( [0, \frac{2}{3}] \) - This range does not include 1 or 2.
4. **Option 4:** Not specified but can be inferred as incorrect based on previous evaluations.

### Conclusion
From our analysis, the only option that includes the values \(1\) and \(2\) is **Option 2: \( \left[\frac{2}{3}, 2\right] \)**.