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Question $6(10$ points$)$

The surface of a certain lake is represented by the region $R$ in the $xy-$plane such that the depth $($in feet$)$ under the point corresponding to $(x,y)$ is given by $f(x,y)=300-2x^2-3y^2.$ Assume that a person in the water is at the point $(4,9).$

a$.$ In what direction should the person swim in order for the depth to increase most rapidly?

b$.$ What is the value of this maximum rate of increase?

c$.$ Find the directional derivative of $f(x,y)$ at the point $(4,9)$ in the direction of the vector vec$(v)=3$hat$(i)-$hat$(j).$