log connection each minute

have you tried the commands by themselves yourself? when i run tmsh show sys connection i get

Really display all connections? (y/n)

which will break your script at this stage

a solution seems to be

tmsh -q show sys connection

although doing a grep doesnt do anything for me here, as the last line is: Total records returned: 119, so it would have to be a grep Total i believe

then the second line, why the -c ?

tmsh show ltm node  | grep 'Current Connections'

works for me fine

then again as above for the final command

script becomes (i have thrown in a date / time stamp, feel free to remove)

 !/bin/sh

 script para ejecutar comando cada minuto

echo "$(date)"
tmsh -q show sys connection | grep Total
tmsh show ltm node node_135.208.45.223 | grep 'Current Connections'
tmsh -q show sys connection | grep Total

crontab looks almost fine although you seem to miss a first

/

root/home/root/Mi_script.sh >> /home/root/conexiones.txt

you could do the >> in the script and not in the crontab, perhaps some linux gurus have opinions on this.