Download Problem 1. Let C > 0. Find all nonnegative differentiable f on [0,+

Problem 1. Let C > 0. Find all nonnegative differentiable f on [0, +∞) which vanish
in 0 and satisfy

Cf (t) |ln f (t)| ,
f (t) > 0,
d
∀t ≥ 0 .
f (t) ≤
0,
dt
f (t) = 0,
Problem 2. Let C > 0. Find all nonnegative differentiable f on [0, +∞) which vanish
in 0 and satisfy the following inequality for all t ≥ 0 and ∈ (0, 1),
d
C
f (t) ≤ f 1− (t) .
dt
Problem 3. Let A ⊂ R. We call a point x ∈ A isolated if there is ρ > 0 with
Bρ (x) ∩ A = {x} (Bρ (x) = y ∈ R |y − x| < ρ = (x − ρ, x + ρ)). If A is closed and
contains no isoated points, then A is called perfect. Prove that every perfect subset A
of R is uncountable.
Problem 4.
(a) Prove the following identity for all n ∈ N \ {0} and x ∈ R:
!
2
n X
k
x(1 − x)
n
−x
xk (1 − x)n−k =
.
n
n
k
k=0
(b) Let f : [0, 1] → R be continuous and define for every n ∈ N \{0} the Polynomial
!
n
X
k
n
Sn (x) =
f
xk (1 − x)n−k .
n
k
k=0
Use (a) to prove that Sn converges uniformly to f on [0, 1].
Problem 5. Find all continuously differentiable functions
u : [0, +∞) × R2 → R,
(t, x, y) 7→ u(t, x, y),
such that

∂u
∂u
∂u


+2
+3
= 0,
∂t
∂x
∂y


− x2 +y 2
u(0, x, y) = e
,
auf [0, +∞) × R2 ,
on R2 .
Problem 6. Consider the map
X : R2 → R2 ,
(x, y) 7→
!
X1 (x, y)
=
X2 (x, y)
1
3
2
!
−x − 4xy
,
−16y 3 − 4x2 y
2
and assume that γ : [0, +∞) → R2 , t 7→ γ(t) = γ1 (t), γ2 (t) is a continuously differentiable curve that, at every t ≥ 0, satisfies the equations

γ10 (t) = X1 γ1 (t), γ2 (t)
γ 0 (t) = X γ (t), γ (t) .
2
1
2
2
Prove that γ [0, +∞) is a bounded subset of R2 .
Problem 7. Let n, k ∈ N \ {0} be given and assume that f : Rn → R is a function,
which is k times differentiable with continuous derivatives of order k. How many
different partial derivatives of order k is it necessary to compute, so to know all the
partial derivatives of order k?
Problem 8. Let U ⊂ Rn (n ≥ 1) be open and bounded and let f : U → R be a
continuous function, which is two times differentiable in U and satisfies the following
equation
n
X
∂2
f (x) < 0
∀x ∈ U .
4f (x) =
2
∂x
i
i=1
Prove that then f takes its minimum on ∂U .
Problem 9. Prove that there is no function u ∈ C 1 [0, +∞) × R which solves

 ∂ u(t, x) + 1 ∂ u(t, x) 2 = 0,
(t > 0, x ∈ R),
∂t
2 ∂x
u(0, x) = e−x2 ,
(x ∈ R).
Problem 10. Let c > 0 and f, g ∈ C 2 ([−1, 1]) with f (±1) = g(±1) = 0. Prove that
there is at most one solution u ∈ C 2 [0, +∞) × [−1, 1] of the system
 2
∂
∂2

u(t, x) = c2 ∂x
(t > 0, x ∈ (−1, 1)),

2 u(t, x),
∂t2



u(t, −1) = u(t, 1) = 0,
(t ≥ 0),
(x ∈ [−1, 1]),

u(0, x) = f (x),




 ∂ u(0, t) = g(x),
∂t
(x ∈ [−1, 1]).
Problem 11. Consider the map
f : [0, +∞) → R,
ˆ
x 7→
0
Is f continuous in 0? Is f differentiable in 0?
Problem 12.
+∞
cos xt
dt.
1 + t2
3
(1) Let k ∈ N \ {0} and assume that U ⊂ Rk is open and closed. Prove then that
either U = ∅ or U = Rk ;
(2) Let k ∈ N \ {0} and u : Rk → Rk be continuously differentiable, with kdukop ≤
L < 1. Prove then that the following map is a diffeomorphism:
Φ : Rk → Rk ,
x 7→ x − u(x) .
Problem 13. Let
F (x, y) = −x4 y 6 + y 4 + 2x3 y 3 − 2y 2 − x2 + 1.
Prove the existence of an interval I ⊂ R with 0 ∈ I and of four different continuously
differentiable functions φi : I → R (i = 1, . . . , 4) with
F x, φi (x) = 0,
∀x ∈ I, i ∈ {1, . . . , 4}.
Problem 14. Prove the following inequality for every n × n–Matrix A:
v
n uX
Y
u n
t
|det A| ≤
|Aij |2 .
j=1
i=1