Not sure when to use "exist" and "exists"

As a non-native English speaker I have never been quite sure how to use the words "exist" and "exists".

For example is the usage of the said words grammatically correct in the following sentences?

1: "there would need to exist a point such that..."
2: "since the point exists, we conclude that..."

In addition it would be helpful if you could point me to some ELI5-like resource, that explains the surrounding grammatical rules thoroughly.

Solution 1:

The verb 'to exist' works in the same way as any other standard English verb.

1. The infinitive follows need as in 'I need to sit down'.

2. 'The point exists' is present tense, third person singular.