Question

The tortoise ran slowly and steadily to win the race.

to win the race -----is the infinitive phrase


I am thinking tortoise ran for what I am saying the phrase is used as noun
Writeacher
"to win the race" is the infinitive phrase, yes.

It's not being used as a noun, though.
Read the explanations and examples here, and re-think your answer.
http://grammar.ccc.commnet.edu/grammar/phrases.htm#infinitive