This video is saying
in order to better visualize the cost function J, I'm going to ...
with the same structure, we could make this sentence
in order to better solve a problem
sounds uncommon, is something wrong?
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It is called a "split infinitive" and there are several previous questions on this topic, such as Splitting the infinitive particle with the verb…is it lawful? Please also see on ELU, Are split infinitives grammatically incorrect, or are they valid constructs? – Weather Vane Feb 20 '20 at 20:46
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@WeatherVane Thanks for your comment. The post on ELU says "Indeed, in many cases, putting the adverbial phrase in the intervening position is the only grammatical place to put it", is it the case in my OP? – brennn Feb 20 '20 at 21:04
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The more normal word order would be "In order to solve a problem better ..." But there are times you don't want to use it. For example, with In order to better understand the almost incomprehensibly convoluted sentence structures in this text ... if you put better after text, it will be too far away from the verb to be easily understood, so you need to put it next to the verb. But for the sentence in the video, the phrase isn't that long, so you could put better after J (I'm not sure whether that's an improvement, though). – Peter Shor Dec 26 '20 at 14:10
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“better” is being used as an adverb here. You can also do the same with “best”.
If X is a better (or the best) solution to problem Y, then you can say that “X better (or best) solves Y”.
Note that X is often implicitly compared to the status quo, such as “we can better serve our customers by staying open an hour later (than we do today).”
The specific structure “to [adverb] [verb]” is called a “split infinitive”, and some purists dislike it, but it is nevertheless commonly used and accepted. The canonical example is “to boldly go” from Star Trek.
StephenS
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"better" is also a verb:
His total of five gold medals is unlikely to be bettered.
You can also find the similar structure in the book of "Interdisciplinarity for the 21st Century" page 113, ( in order to better solve it)
lee
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The fact that better is also a verb is completely irrelevant to this question, and a distraction. – Colin Fine Nov 24 '20 at 18:13
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1@ColinFine Now that I am sleeping on it well, it seems that "better" is an adverb here. Am I right? – lee Nov 24 '20 at 21:56