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For an academic study, I am planning to find the effect of the notes at which the melody changes its direction.
So, for example in a melody like: C3 G3 F3 D3 E3
G3 and D3 are those notes that I am interested in, because the notes next to G3 are both lower than it, and similarly, the notes next to D3 are both higher than it.
So, is there a name for such notes, for example elbow note maybe?
In most theories of music, the concept doesn't have enough value to be worth naming. Of course if your research shows that the concept does have some value, you can therefore name it yourself!
On a short timescale (two or three notes), the term Cambiata is used, especially in polyphonic music.
On a long timescale, terms like "climax of a phrase" (or of a piece) express the general idea, but may include other aspects of the music, such as dynamics, texture, or instrumentation.
Some possibilities:
We also talk about "extremes of range," so perhaps you could discuss "the local extreme of G3." The problem here is that clarifying "the local upper extreme" starts to be linguistically cumbersome.
And since you're focusing on pitches "at which the melody changes its direction," maybe you should use terminology that emphasizes that instead?
For notes that aren't part of the underlying chord progression, https://en.m.wikipedia.org/wiki/Nonchord_tone has names of several types of "elbow notes", such as Neighbouring Tone (approach and leave by step) and Appoggiatura (approach by leap, leave by step, accent the "elbow note").
The above doesn't 100% answer the question, but it's possible that music theorists generally aren't interested in the generalized "elbow note" because of ubiquity problems like Tim mentioned.
Toch ("The Shaping Forces of Music") describes melodies as being wave-like with successions of higher and lower waves. This suggest the terms "peak" and "trough" for such notes. I think I have seen the terms "highs and lows" used, but that may be overworked.
In general you wouldn't single out these pitches, but they would be apart of the contour or shape of the melody. There's a lot of detail you can get based on the contour of the piece and the basic contours you can have are:
Ascending
First pitch lower than middle pitch. Middle pitch lower than last pitch.
Descending
First pitch higher than middle pitch. Middle pitch higher than final pitch.
Concave
First and last pitch higher than middle pitch.
Convex
First and last pitch lower than middle pitch.
Horizontal
First, middle, and last pitch, roughly equivalent in range.
As you break down the contour into bigger or smaller pieces these notes will stand out as will the lower notes on the other end of the spectrum, but there's no reason to call them anything fancy. The reason why is the contour itself is much more important the individual notes that make it up. The most you'll ever see them called is a climax and if you're really specific a local climax.
I don't think there is a universal term for this, though Philip Ball uses the term "reversal" in his book "The Music Instinct".
If you were wanting to break new ground, here are two suggestions: 1.For medievalists, perhaps acme localis 2.For cutting edge-ists there is (as of now I think) the nonce word locacme, derived from 1.
