An egg will break when dropped from a great enough height. Specifically, in an n$-$story building, there must be an integer f$,$ called the critical floor, such that an egg dropped from the f$-$th floor breaks, but an egg dropped from the f $-1-$th floor does not. By convention, if an gg does not break when dropped from any floor, we write f $=$ n $+1.$ The only operation available to you is to drop an egg from any floor to see if it breaks $($this is called an egg$-$drop$).$ Broken eggs cannot be reused. Let E$($k$,$ n$)$ denote the minimum number of egg$-$drops needed to determine the critical floor given k eggs. $1.$ Prove that E$(1,$ n$)=$ n$.2.(5$ Points Extra Credit$)$ Show that E$($k$,$ n$)=0($n$/$k$).$ Hint $1$ Use recursion $$ Hint $2$ Use mathematical induction on k $$ Hint $3$ Treat in and k as real numbers and find the minimum by differentiating $($taking the derivative of$)$ Ek$,$ n$).3.$ Give a recurrence for E$($k$,$ n$).$ What is the runtime of a dynamic programming algorithm that outputs E$($k$,$ n$)?$ Note This is different from part $2,$ since k$,$ n$,$ and E$($k$,$ n$)$ must be integers.