If the point `(alpha, alpha^(4))` lies on or inside the triangle formed by lines `x^(2)y+xy^(2)-2xy=0`, then the largest value of `alpha` is .

Answer

To solve the problem, we need to determine the largest value of \( \alpha \) such that the point \( (\alpha, \alpha^4) \) lies on or inside the triangle formed by the lines given by the equation \( x^2y + xy^2 - 2xy = 0 \).

### Step-by-Step Solution:

1. **Rewrite the Equation**:
The given equation is \( x^2y + xy^2 - 2xy = 0 \). We can factor this equation:
\[ xy(x + y - 2) = 0 \]
This gives us three cases:
- \( x = 0 \)
- \( y = 0 \)
- \( x + y - 2 = 0 \)

2. **Identify the Lines**:
The lines represented by the factors are:
- \( x = 0 \) (the y-axis)
- \( y = 0 \) (the x-axis)
- \( x + y = 2 \) (a line with intercepts at (2, 0) and (0, 2))

3. **Determine the Vertices of the Triangle**:
The triangle formed by these lines has vertices at:
- \( (0, 0) \) (intersection of \( x = 0 \) and \( y = 0 \))
- \( (2, 0) \) (intersection of \( y = 0 \) and \( x + y = 2 \))
- \( (0, 2) \) (intersection of \( x = 0 \) and \( x + y = 2 \))

4. **Check if the Point Lies Inside the Triangle**:
We need to check if the point \( (\alpha, \alpha^4) \) lies on or inside the triangle. This can be done by substituting the coordinates into the inequalities derived from the lines.

5. **Inequality from the Line \( x + y = 2 \)**:
For the line \( x + y - 2 \geq 0 \):
\[ \alpha + \alpha^4 - 2 \geq 0 \]
Rearranging gives:
\[ \alpha^4 + \alpha - 2 \geq 0 \]

6. **Finding Roots of the Polynomial**:
We need to find the values of \( \alpha \) for which \( \alpha^4 + \alpha - 2 = 0 \). We can check some values:
- For \( \alpha = 1 \):
\[ 1^4 + 1 - 2 = 0 \quad \text{(satisfies)} \]
- For \( \alpha = 2 \):
\[ 2^4 + 2 - 2 = 16 + 2 - 2 = 16 \quad \text{(does not satisfy)} \]

7. **Analyze the Behavior of the Polynomial**:
The polynomial \( \alpha^4 + \alpha - 2 \) is continuous and increases for \( \alpha > 1 \). Since it equals zero at \( \alpha = 1 \) and is positive for values greater than 1, we check if it becomes negative before reaching 2.

8. **Conclusion**:
The largest value of \( \alpha \) for which \( \alpha^4 + \alpha - 2 \geq 0 \) is \( \alpha = 1 \).

### Final Answer:
The largest value of \( \alpha \) such that the point \( (\alpha, \alpha^4) \) lies on or inside the triangle is:
\[ \boxed{1} \]