Consider a government$-$controlled firm that produces output y according to the pro$-$ duction function y $=$ K $^$ a $*$ L where a $>1$ is a parameter. Suppose the firm is directed to maximize output subject to meeting the cost constraint $2$ overline K $+$ L $=12$

$($a$)$ Set up the Langragian

$($b$)$ Write the FOC

$($c$)$ Find L $^*$ K $^*$ and

lambda $^*$

$($d$)$ Use the Bordered Hessian Matrix to check the SOCS. Is that truly a maximum?

A$)$ The candidate solution is a local minimum,

B$)$ The candidate solution is a local maximum

C$)$ The candidate solution is a global maximum

D$)$ The candidate solution is a global minimum

E$)$ None of the above

$($e$)$ Find the maximal output y $^*($i$.$e$.$ the maximum$-$value function$)$

$($f$)$ Use the envelope theorem to find how the maximal output y $^*$ changes as the parameter a varies.

Please do out full workings and explanations.