Description
EIGHTH HOMEWORK
1. (10 pts) (4.2.1)
2. (5 pts) (4.2.6)
3. (5 pts) (4.2.8)
4. (10 pts) (4.2.22)
5. (10 pts) (4.2.23)
6. (5 pts) (4.2.33)
7. (5 pts) (4.2.46(b))
8. (20 pts) We will show in this problem that amongst all boxes with
surface area a, the volume is a maximum if and only if the box is a
cube. Let
V : R3 −→ R,
be the function V (x, y, z) = xyz. Then we want to maximise V on the
set A of all points where x > 0, y > 0, z > 0 and 2(xy + yz + zx) = a.
(i) Show that there is a unique point P ∈ A where V has a constrained
critical point.
Let K ⊂ A be the subset of points (x, y, z) where
√a √a √a
x ≥ 3
√6 y ≥ 3
√6
and z ≥ 3
√6
.
(ii) Show that if Q ∈ A − K (so that Q is in A but not K) then
V (Q) < V (P).
(iii) Show that K is compact.
(iv) Show that V has a maximum on K at P.
(v) Show that V has a maximum on A at P.
8. (10 pts) (4.3.2)
9. (10 pts) (4.3.8)
10. (10 pts) (4.3.18)
Just for fun: What is the value of the limit:
sin(tan x) − tan(sin x) lim . x 0 sin−1
(tan−1(x)) − tan−1(sin−1 → (x))
1
18.022 Calculus of Several Variables
