3. If f(x, y)=5 − 4x3 + 8y2, find a unit vector ~u for which the directional derivative

D~uf(1, 1) is zero.

im not sure how to do this one

i figured this one out!

To find a unit vector ~u for which the directional derivative D~uf(1, 1) is zero, we need to follow a few steps.

Step 1: Find the gradient vector ∇f(x, y) of the function f(x, y).
The gradient vector is given by ∇f(x, y) = (∂f/∂x, ∂f/∂y). In this case:
∂f/∂x = -12x^2
∂f/∂y = 16y

So, the gradient vector is ∇f(x, y) = (-12x^2, 16y).

Step 2: Evaluate the gradient vector at the given point (1, 1).
Substituting x = 1 and y = 1 into the gradient vector, we get:
∇f(1, 1) = (-12(1)^2, 16(1)) = (-12, 16).

Step 3: Find the magnitude of the gradient vector at the given point.
The magnitude of the gradient vector is given by ||∇f(x, y)|| = sqrt((∂f/∂x)^2 + (∂f/∂y)^2).
Substituting x = 1 and y = 1 into the magnitude formula, we get:
||∇f(1, 1)|| = sqrt((-12)^2 + (16)^2) = sqrt(144 + 256) = sqrt(400) = 20.

Step 4: Divide the gradient vector by its magnitude to get the unit vector ~u.
The unit vector ~u is obtained by dividing each component of the gradient vector by the magnitude:
~u = (∇f(1, 1)) / ||∇f(1, 1)|| = (-12/20, 16/20) = (-0.6, 0.8).

So, the unit vector ~u for which the directional derivative D~uf(1, 1) is zero is (-0.6, 0.8).