Question

Are the following statements true or false?


1. If u⃗
is a unit vector, then fu⃗ (a,b)
is a vector.



2. ∇f(a,b)
is a vector in 3-dimensional space.



3. Suppose fx(a,b)
and fy(a,b)
both exist. Then there is always a direction in which the rate of change of f
at (a,b)
is zero.



4. If u⃗
is perpendicular to ∇f(a,b)
, then fu⃗ (a,b)=⟨0,0⟩
.



5. The gradient vector ∇f(a,b)
is tangent to the contour of f
at (a,b)
.


6. If f(x,y)
has fx(a,b)=0
and fy(a,b)=0
at the point (a,b)
, then f
is constant everywhere.



7. fu⃗ (a,b)
is parallel to u⃗
.



8. fu⃗ (a,b)=||∇f(a,b)||

Answer 1

1. False
2. True
3. False
4. True
5. False
6. True
7. True
8. False