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Question
A hiker walks 12km in one hour. Prove that there exists an interval of time of a half-hour during which he walks exactly 6km.
Define an application
from
to
by
is the walked distance, in kilometers, at time
Define then an application
from
to
by
Prove then that
belongs to the segment
and use the theorem of mean values.
We denote by
the application from
to
such that
is the walked distance at time
expressed in kilometers and by
the application defined from
to
by:
We infer that
The middle of segment
hence
We assume that application
is continuous, we infer, with classical theorems, that application
is continuous, therefore by the intermediate value theorem, there exists
such that
which gives us
hence the hiker walked exactly 6km between time
and time
