For what value of k does the equation, 2x2  - 5x + k = 0 have both the roots in the interval (0, 1)?

The roots of the quadratic equation ax^2 + bx + c = 0 are given by the formula x = [-b ± sqrt(b^2 - 4ac)] / 2a.

For the roots to lie in the interval (0,1), the discriminant (b^2 - 4ac) must be greater than or equal to 0.

In this case, a = 2, b = -5, and c = k.

So, we have (-5)^2 - 42k >= 0, which simplifies to 25 - 8k >= 0.

Solving for k, we get k <= 25/8 = 3.125.

However, this is not the final answer because we also need to consider the fact that the roots must lie in the interval (0,1).

The roots of the equation are given by x = [5 ± sqrt(25 - 8k)] / 4.

For the roots to lie in the interval (0,1), we must have 0 < [5 - sqrt(25 - 8k)] / 4 < 1 and 0 < [5 + sqrt(25 - 8k)] / 4 < 1.

Solving these inequalities, we get 3 < k < 3.125.

Therefore, the value of k for which the equation 2x^2 - 5x + k = 0 has both roots in the interval (0,1) is k in the interval (3, 3.125).