For what value of \( \Large \lambda \) the sum of the squares of the roots of \( \Large x^{2}+ \left(2+\lambda\right)n-\frac{1}{2} \left(1+\lambda\right)=0 \) is minimum?

Given equation is \( \Large x^{2}+ \left(2+\lambda\right)x-\frac{1}{2} \left(1+\lambda \right)=0 \).

Let \( \Large \alpha \  and \ \beta \) are the roots of given equations

=> \( \Large \alpha + \beta =- \left(2+ \lambda \right) \) and \( \Large \alpha \beta = - \left(\frac{1+\lambda}{2}\right) \)

Now, \( \Large \alpha ^{2}+ \beta ^{2}= \left( \alpha + \beta \right)^{2}-2 \alpha \beta \)

=> \( \Large \alpha ^{2}+ \beta ^{2}= \left[ - \left(2+\lambda\right)^{2}+2\frac{ \left(1+\lambda\right) }{2} \right] \)

=>\( \Large \alpha ^{2}+ \beta ^{2}= \lambda^{2}+4+4\lambda+1+\lambda=\lambda^{2}+5\lambda+5 \)

Now we take the option simultaneously

=> It is minimum for \( \Large \lambda = \frac{1}{2} \).