a and b are the roots of the quadratic equation `x^2 + lambdax - 1/(2lambda^(2)) =0` where `x` is the unknown and `lambda` is a real parameter. The minimum value of `a^(4) + b^(4)` is:

Answer

To find the minimum value of \( a^4 + b^4 \) where \( a \) and \( b \) are the roots of the quadratic equation

\[ x^2 + \lambda x - \frac{1}{2\lambda^2} = 0, \]

we can follow these steps:

### Step 1: Identify the sum and product of the roots
From Vieta's formulas, for the quadratic equation \( x^2 + bx + c = 0 \):
- The sum of the roots \( a + b = -\lambda \)
- The product of the roots \( ab = -\frac{1}{2\lambda^2} \)

### Step 2: Express \( a^4 + b^4 \) in terms of \( a + b \) and \( ab \)
We can use the identity:

\[ a^4 + b^4 = (a^2 + b^2)^2 - 2(a^2b^2) \]

To express \( a^2 + b^2 \) and \( a^2b^2 \):
- We know \( a^2 + b^2 = (a + b)^2 - 2ab \)
- And \( a^2b^2 = (ab)^2 \)

Substituting the values we have:

\[ a^2 + b^2 = (-\lambda)^2 - 2\left(-\frac{1}{2\lambda^2}\right) = \lambda^2 + \frac{1}{\lambda^2} \]

And

\[ a^2b^2 = \left(-\frac{1}{2\lambda^2}\right)^2 = \frac{1}{4\lambda^4} \]

### Step 3: Substitute back into the expression for \( a^4 + b^4 \)
Now substituting these into the expression for \( a^4 + b^4 \):

\[ a^4 + b^4 = \left(\lambda^2 + \frac{1}{\lambda^2}\right)^2 - 2\left(\frac{1}{4\lambda^4}\right) \]

Expanding this:

\[ = \lambda^4 + 2 + \frac{1}{\lambda^4} - \frac{1}{2\lambda^4} \]

Combining the terms gives:

\[ = \lambda^4 + 2 + \frac{1}{2\lambda^4} \]

### Step 4: Find the minimum value
Let \( F(\lambda) = \lambda^4 + 2 + \frac{1}{2\lambda^4} \).

To find the minimum, we differentiate \( F \) with respect to \( \lambda \):

\[ F'(\lambda) = 4\lambda^3 - \frac{2}{\lambda^5} \]

Setting \( F'(\lambda) = 0 \):

\[ 4\lambda^3 = \frac{2}{\lambda^5} \]

Multiplying both sides by \( \lambda^5 \):

\[ 4\lambda^8 = 2 \implies \lambda^8 = \frac{1}{2} \implies \lambda = \left(\frac{1}{2}\right)^{1/8} \]

### Step 5: Evaluate \( F \) at the critical point
Now substituting \( \lambda = \left(\frac{1}{2}\right)^{1/8} \) back into \( F \):

\[ F\left(\left(\frac{1}{2}\right)^{1/8}\right) = \left(\left(\frac{1}{2}\right)^{1/8}\right)^4 + 2 + \frac{1}{2\left(\left(\frac{1}{2}\right)^{1/8}\right)^4} \]

Calculating each term:

1. \( \left(\frac{1}{2}\right)^{1/2} = \frac{1}{\sqrt{2}} \)
2. \( \frac{1}{2\left(\frac{1}{2}\right)^{1/2}} = \frac{1}{2 \cdot \frac{1}{\sqrt{2}}} = \frac{\sqrt{2}}{2} \)

Thus,

\[ F = \frac{1}{\sqrt{2}} + 2 + \frac{\sqrt{2}}{2} \]

Combine terms to find the minimum value:

\[ = 2 + \frac{1 + \sqrt{2}}{\sqrt{2}} = 2 + \sqrt{2} \]

### Final Answer
The minimum value of \( a^4 + b^4 \) is

\[ \boxed{2 + \sqrt{2}}. \]