Let the set of all positive values of $\lambda$,for which the point of local minimum of the function $f(x) = 1 + x(\lambda^2 - x^2)$ satisfies $\frac{x^2+x+2}{x^2+5x+6} < 0$,be $(\alpha, \beta)$. Then $\alpha^2 + \beta^2$ is equal to:

Find the absolute maximum of $x^{40}-x^{20}$ on the interval $[0,1]$.

Find the maximum and the minimum values,if any,of the function $f$ given by $f(x) = x^{2}, x \in R$.

Find the local maximum and local minimum values for the function $h(x) = \sin x + \cos x$ in the interval $0 < x < \frac{\pi}{2}$.

Let $f :[2,4] \rightarrow R$ be a differentiable function such that $(x \ln x) f'(x) + (\ln x + 1) f(x) \geq 1$ for all $x \in [2,4]$,with $f(2) = \frac{1}{2}$ and $f(4) = \frac{1}{4}$. Consider the following two statements:
$(A): f(x) \leq 1$ for all $x \in [2,4]$
$(B): f(x) \geq \frac{1}{8}$ for all $x \in [2,4]$
Then,

The minimum value of $e^{(2x^2 - 2x + 1)\sin^2 x}$ is