Mathematics - Application of Derivative Question with Solution | TestHub

$\begin{array}{rl}& \frac{x^2+x+2}{x^2+5x+6}<0\\ & \Rightarrow \frac{1}{(x+2)(x+3)}<0\end{array}$
$\begin{array}{rl}& x\in (-3,-2)\dots \dots \dots ..(1)\\ & f(x)=1+x\left({\lambda }^2-x^2\right)\end{array}$
Finding local minima $f^{\prime }(x)=\left({\lambda }^2-x^2\right)+(-2x)\cdot x$ Put ${\mathrm{f}}^{\prime }(\mathrm{x})=0$ $\begin{array}{rl}& \Rightarrow {\lambda }^2=3x^2\\ & \Rightarrow x=\pm \frac{\lambda }{\sqrt{3}}\end{array}$We want local min $\Rightarrow \mathrm{x}=\frac{-\lambda }{\sqrt{3}}$ from (1) $x\in (-3,-2)$ $\begin{array}{rl}& -3<\frac{-\lambda }{\sqrt{3}}<-2\\ & 3\sqrt{3}>\lambda >2\sqrt{3}\\ & \alpha =2\sqrt{3},\beta =3\sqrt{3}\\ & {\alpha }^2+{\beta }^2=12+27=39\end{array}$