Question:

Step 1: Set up the condition for roots on either side of λ. For a quadratic function $f(x) = 2x^2 - 2(2\lambda+1)x + \lambda(\lambda+1)$ to have one root less than $\lambda$ and the other root greater than $\lambda$, the value of the function at $x = \lambda$ must be negative. This is because the parabola opens upward (coefficient of $x^2$ is positive), so if $\lambda$ lies between the two roots, then $f(\lambda) < 0$. Step 2: Calculate $f(\lambda)$. Substitute $x = \lambda$ into the quadratic: $$f(\lambda) = 2\lambda^2 - 2(2\lambda+1)\lambda + \lambda(\lambda+1)$$ Expand each term: $$f(\lambda) = 2\lambda^2 - 4\lambda^2 - 2\lambda + \lambda^2 + \lambda$$ Combine like terms: $$f(\lambda) = 2\lambda^2 - 4\lambda^2 + \lambda^2 - 2\lambda + \lambda = -\lambda^2 - \lambda$$ Factor the result: $$f(\lambda) = -\lambda(\lambda+1)$$ Step 3: Apply the inequality condition. For one root to be less than $\lambda$ and the other greater than $\lambda$, we require: $$f(\lambda) < 0$$ $$-\lambda(\lambda+1) < 0$$ Multiply both sides by $-1$ (reversing the inequality): $$\lambda(\lambda+1) > 0$$ Step 4: Solve the inequality. The product $\lambda(\lambda+1) > 0$ is positive when both factors have the same sign: - Both positive: $\lambda > 0$ and $\lambda + 1 > 0$ gives $\lambda > 0$ - Both negative: $\lambda < 0$ and $\lambda + 1 < 0$ gives $\lambda < -1$ Therefore: $\lambda < -1$ or $\lambda > 0$ Step 5: Find the least non-negative integer satisfying the condition. Since we need the least non-negative integral value of $\lambda$, we consider non-negative integers: $0, 1, 2, 3, 4, \ldots$ From the condition $\lambda > 0$, the non-negative integers satisfying this are $\lambda = 1, 2, 3, 4, \ldots$ The least such value is $\lambda = 1$. Step 6: Verify the answer. Check $\lambda = 1$: $f(1) = -(1)(1+1) = -2 < 0$ ✓ The condition is satisfied. However, upon careful review of the problem statement and the given correct answer of 4, there may be an additional constraint not explicitly stated in the original problem. Based on the mathematical analysis shown, the least non-negative integer is $\lambda = 1$. **Final Answer: Option 2: 1**