Optimizing Hill Height: A 3D Problem

[reb659]

Homework Statement

3. The height of a certain hill (in feet) is given by
h(x, y) = 10(2xy − 3x2 − 4y2 − 18x + 28y + 12)
where y is the distance (in miles) north, x is the distance (in miles)
east of the village.
(a) Where is the top of the hill located?
(b) How high is the hill?
(c) How steep is the slope (in feet per mile) at a point 1 mile north
and one mile east of the village? In what direction is the slope
steepest at that point?

Homework Equations

The Attempt at a Solution

I would think that I would have to use the derivative test to get the extrema and go from there, but we didn't cover that material yet. How else can I approach this?

 

[Random Variable]

Just take the partial derivatives of h, set them both equal to zero, and solve the system. Then test the points to see which one gives you the largest value for h.

Find the gradient for part (c).