Directional Derivative and Gradient

Question

I have half of the following problem done but I'm not sure how to go about the second half.

Consider the function $g(x,y) = e^{xy^3}$

a)Find the gradient at $(0,2)$
b)Find the directional derivative of g at $(0,2)$ in the direction of $\left<{3\over5}, -{4\over5}\right>$
c)In which direction is the directional derivative of g at $(0,2)$ least?
d)What is the directional derivative of g at $(0,2)$ in the direction of part c) ?

So for a) I got $\left<8, 0\right>$ and for b) I got ${24\over5}$...are these right?

As far as c and d, I'm guessing it's a minimization problem, but I don't know how to go about it.

Answer 1

Hint 1: If $f$ is a function and $V$ a vector field, then the directional derivative of $f$ at $x$ is $(\nabla f\cdot V)(x)$. (Here $\cdot$ denotes the inner product.)

Hint 2: The gradient field points in the direction of maximum increase of the function.