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" I was a fool to believe her. "
Is the above to+infinitive structure similar to the one in "I was the first man to win the championship"
where the to+infinitive structure is equivalent to an adjective clause ?
Comments  
They look the same to me.
Here are some ways of analyzing your sentences.

I was a fool to believe her.

Infinitive with adverbial function. Answers the question "Why?" I was a fool because I believed her.

I was the first man to win the championship.

Infinitive with adjectival function. I was the first among the men who would win the championship.

CJ
Students: We have free audio pronunciation exercises.
(1) I respectfully agree that the infinitive phrase is adjectival and modifies "fool."

(2) Some grammarians suggest that adjectival infinitives are just a short way to express an adjectival/relative clause.

(3) Possibly some people would suggest that the "full" sentence might be something like: I was a fool who believed her.

(4) One authority (who, of course, could be wrong) has suggested a test to determine whether an infinitive phrase is adverbial or not. He says to put "in order" in front of "to." If it makes sense, then the infinitive is adverbial.

(a) I was a fool in order to believe her.

(i) In my opinion, this does not make sense. It seems to imply that I purposely became a fool so that I would be able to believe her.

(ii) Compare: I played the fool to trick her out of her money. = I played the fool in order to trick her out of her money. That is, I think, obviously adverbial.

(5) As we all know, sometimes there are several ways to analyze a sentence. Sometimes there is no single "right" answer.
AnonymousOne authority (who, of course, could be wrong) has suggested a test to determine whether an infinitive phrase is adverbial or not. He says to put "in order" in front of "to." If it makes sense, then the infinitive is adverbial.
You may recall from logic class that this can be true without implying that if it doesn't make sense then the infinitive is not adverbial. It's just one way to test if the infinitive is adverbial. Another is to see if the paraphrase "for having ..." makes sense.

I was a fool to believe her. = I was a fool for having believed her.
Jake was right to complain about it. = Jake was right for having complained about it.
Karen was crazy to try that. = Karen was crazy for having tried that.
AnonymousPossibly some people would suggest that the "full" sentence might be something like: I was a fool who believed her.
That would be an unfortunate suggestion because that's not the meaning of the original sentence. The idea is that the speaker felt like a fool because he had believed her. It's not that he was a certain kind of fool -- one of those who believed her. The story would be different if the sentence had said, "She needed to find a fool to believe her", but that's not the idea behind the original sentence.
AnonymousSometimes there is no single "right" answer.
And sometimes there are a good many wrong answers. Emotion: sad

CJ
Thank you for your insightful answer. You have certainly given us all food for thought. I shall study your arguments carefully. And continue trying to find other opinions. Hopefully, other people will join this thread.
Students: Are you brave enough to let our tutors analyse your pronunciation?
Debpriya De:

(1) I have researched your intriguing question some more and hereby present my conclusions for your consideration:

(a) Mr. CalifJim is correct: "to believe her" is being used in an adverbial sense.

(b) Here are some paraphrases that I have been able to find (the first two are Mr. CalifJim's):

(i) I was a fool because I believed her.

(ii) I was a fool for having believed her.

(iii) I was a fool for believing her.

(iv) I was a fool if I believed her.

(v) I was a fool in respect of believing her.

(c) I found a difference of opinion as to what "to believe her" modifies.

(i) Some believe that it simply modifies the verb "was," since it is an infinitive clause being used adverbially as a clause of cause/reason.

(ii) Some feel it more accurate to say it modifies the whole sentence.

(iii) The most important point, however, is that it does NOT modify only "fool."