The value of λ such that sum of the squares of the roots of the quadratic equation, x 2 + 3 - λ…
The value of such that sum of the squares of the roots of the quadratic equation, has the least value is:
Solution
We have, $x^{2}+(3-\lambda)x+2=\lambda$, have roots $\alpha$ & $\beta$.
Then, $f(\lambda)=\alpha^{2}+\beta^{2}$
$=(\alpha+\beta)^{2}-2\alpha\beta=(3-\lambda)^{2}-2(2-\lambda)=\lambda^{2}-6\lambda+9-4+2\lambda=\lambda^{2}-4\lambda+5$
$f'(\lambda)=2\lambda-4=0$ for the minimum value of $f(\lambda)$
As $f''(\lambda)=2 > 0$
$\Rightarrow \lambda=2$