Q$3(40$ pts$).$

a$.(10$ pts$)$ Prove that if $T(n)=8n^2+2n+15,$ then $T(n)=O(n^2).$ Hint: You need to find two

positive constant values $c$ and $n_0$ such that $T(n)c{n}^2$ for all $n{n}_0.$

b$.(10$ pts$)$ Suppose the running time of an algorithm is $7n^2+22m^2+4nm+10m+200,$

where $n$ and $m$ are positive integers such that $nm.$ Prove that this algorithm is $O(n^2).$

c$.(20$ pts$)$ Give an analysis of the running time $($Big$-$Oh notation$)$ for the following

algorithms: