$($B$1)[2$ pts each$]$ An airline sells two types of tickets for a particular flight, business$-$class and first$-$class. If $x$ represents the number of business$-$class tickets sold and $y$ represents the number of first$-$class tickets sold, then the profit $($in hundreds of dollars$)$ is given by the equation

$P(x,y)=20x+35y-\frac{1}{40}{x}^2-\frac{1}{15}xy-\frac{1}{10}{y}^2-800.$

a$.$ Find the critical points of $P(x,y).$

b$.$ Use the second derivative test to determine whether the point or points you found in part a are minimum or maximum values of $P.$

c$.$ Evaluate $P$ at the point or points you found in part a$.$

d$.$ Use your answers in parts $b$ and $c$ to write a sentence indicating the maximum profit $($in dollars$)$ that the airline can make on this flight and how many tickets of each type they must sell to achieve this maximum profit.