- Warm-up-exercises
Find a zero for the function
-
in the interval
using the interval bisection method with a maximum error of
.
Let
-
be a
continuous function.
Show that
is not
surjective.
Give an example of a bounded interval
and a continuous function
-
such that the image of
is bounded, but the function admits no maximum.
Let
-
be a continuous function. Show that there exists a continuous extension
-
of
.
Let
-
be a continuous function defined over a real interval. The function has at points
,
,
local maxima. Prove that the function has between
and
has at least one local minimum.
Determine directly, for which
the power function
-
has an extremum at the point zero.
Show that the Intermediate value theorem for continuous functions from
to
does not hold.
Determine the limit of the sequence
-
- Hand-in-exercises
Determine the minimum of the function
-
Find for the function
-
a zero in the interval
using the interval bisection method, with a maximum error of
.
Determine the limit of the sequence
-
The next task uses the notion of an even and an odd function.
Let
-
be a
continuous function.
Show that one can write
-
![{\displaystyle {}f=g+h\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32ae3382cd89006862eae9fa5dd5d4aa98c1493f)
with a continuous
even function
and a continuous
odd function
.
The following task uses the notion of fixed point.
Let
-
be a continuous function from the interval
into itself. Prove that
has a fixed point.