At (2,1), I have to find the direction where $\displaystyle f(x,y)=x^2+3xy-y^2$ is increasing fastest.

So I did:

$\displaystyle \triangledown f(2,1)= \langle 2(2)+3, 3(2)-2 \rangle = \langle 7,4 \rangle$

Next I have to find the rate of growth in this direction, so I did:

$\displaystyle ||\triangledown f(2,1)|| = \sqrt{7^2+4^2} = \sqrt{65}$

Finally I have to find the direction a this point where f is changing "least rapidly." Not sure what to do here...