Problem 1. Let f ∈ C ∞ ([−1, 1]) with f (n) (x) ≥ 0 for all x ∈ [0, 1] and all n ∈ N.
Show that f is analytic on ]0, 1[. Is f necessarily analytic on the whole interval ]−1, 1[?
Problem 2. Let α > β > 0. Consider the function f : [0, +∞[→ R given by
(
cos αx−cos βx
, x > 0,
x
f (x) =
0
, x = 0.
´R
Zeige, dass f |[0,R] für jedes R > 0 eine Regelfunktion ist, und berechne limR%+∞ 0 f (x) dx.
ˆ ∞
xα cos (ex ) dx exist?
Problem 3. For which real numbers α does the integral
1
Problem 4. Show that every open set U ⊂ Rn is the union of countably many cubes
of the form (x1 , x1 + h) × (x2 , x2 + h) × · · · × (xn , xn + h).
Problem 5. Let B ⊂ Rn be the closed unit ball. Let {fn }n∈N , fn : B → R be a
sequence of continuous functions. Prove that, eithere there is x ∈ B with fn (x) = 0
∀n ∈ N, or there is N ∈ N such that the system


 f1 (x) = 0
..
.


fN (x) = 0
does not have any solution.
Problem 6.
(1) Let p be a polynomial in one real variable such that p(x) > 0 for every x ∈ R.
Prove that p attains its minimum value.
(2) Show the existence of a polynomial of two variables which is everywhere positive
on R2 but does not attain its minimum.
Problem 7.
(1) Let K = [−1, 1]2 and f : R2 → R differentiable. Prove or disprove the following
assertion: If
f (−1, −1) = f (−1, 1) = f (1, −1) = f (1, 1) = 0 ,
then there is x ∈] − 1, 1[2 such that
∇f (x) = 0 .
(2) Give a generalization to Rn of the Rolle’s Theorem.
1
2
Problem 8.
(1) Let f ∈ C 1 (R; R). Assume that f has two local minimal at the points x1 6= x2 ∈
R. Prove that f has at least a third critical point, i.e.
∃ x3 6∈ {x1 , x2 }
such that
∇f (x3 ) = 0 .
(2) Does the same conclusion hold when f ∈ C 1 (R2 ; R)?
Problem 9.
(1) Let f ∈ C 1 (R; R) with f 0 (x) 6= 0 for all x ∈ R. Prove that
(a) the image imf of f is an open subset of R;
(b) f , as map from R to imf , is globally invertible.
(2) Does the same conclusion hold for C 1 maps from R2 to R2 ?