Gradient and Directional derivative

Let f (x, y) = x²y. Then ∇f(x, y) = [x², 2xy]^t.

• True

• False

The gradient of f (x, y) is a scalar value.

• True

• False

A directional derivative represents a function's rate of change in any given direction.

• True

• False

A directional derivative is a vector.

• True

• False

The direction of the greatest decrease of f(x, y) is -∇f(x, y).

• True

• False

• 6(2x²+y)⁵

• (4x+y)⁶

• 6(2x²+y)⁵⋅4x

• None of the above

Let f (x, y) = x²y. Find ∇f(1,2).

• [1, 4]^t

• [1, -4]^t

• [4, -1]^t

• [4, 1]^t

If f(x, y, z) = e^xyz. Find the direction of the maximum rate of change of f at (0, 1, -1).

• [0, -1, 0]^t

• [-1, 0, 0]^t

• [0, 0, -1]^t

• None of the above

• None of the above

• 1*(3/5) + 4*(4/5) = 19/5 = 3.8

• 1*3 + 4*4 = 19

• 4*(3/5) + 1*(4/5) = 16/5 = 3.2

• 4*3 + 1*4 = 16