The two positive numbers with sum $t$,and the sum of their squares is minimum are

Let $\alpha = \sum_{k=1}^{\infty} \sin^{2k}\left(\frac{\pi}{6}\right)$. Let $g:[0,1] \rightarrow \mathbb{R}$ be the function defined by $g(x) = 2^{\alpha x} + 2^{\alpha(1-x)}$. Then,which of the following statements is/are $TRUE$?
$(A)$ The minimum value of $g(x)$ is $2^{7/6}$
$(B)$ The maximum value of $g(x)$ is $1 + 2^{1/3}$
$(C)$ The function $g(x)$ attains its maximum at more than one point
$(D)$ The function $g(x)$ attains its minimum at more than one point

If $x$ and $y$ are two positive integers such that $x+y=24$ and $x^3 y^5$ is maximum, then $x^2+y^2=$

$A$ stone moving vertically upwards has its equation of motion $s = 490t - 4.9t^2$. The maximum height reached by the stone is

The maximum value of the function $f(x)=e^x+x \ln x$ on the interval $1 \leq x \leq 2$ is

Local maximum and local minimum values respectively of the function $f(x)=(x-1)(x+2)^2$ are