Explanation

(c)
Given equation is $x^{2} + (2 + \lambda)x - \frac{1}{2}(1 + \lambda) = 0$.
Let $\alpha\ and\ \beta$ are the roots of the given equation.
$\Rightarrow \ \alpha + \beta = - (2 + \lambda)$ and $\alpha\beta = - \left( \ \frac{1 + \lambda}{2} \right)$
Now, $\alpha^{2} + \beta^{2} = (\alpha + \beta)^{2} - 2\alpha\beta$
$\Rightarrow \ \alpha^{2} + \beta^{2} = \left\lbrack - (2 + \lambda) \right\rbrack^{2} + 2$ $\frac{(1 + \lambda)}{2}$
$\Rightarrow \ \alpha^{2} + \beta^{2} = \lambda^{2} + 4 + 4\lambda + 1 + \lambda = \lambda^{2} + 5\lambda + 5$
Now, we take the option simultaneously.
$\Rightarrow \$ It is minimum for $\lambda = \frac{1}{2}$.