#tangency

Intuition: We know by our definition that the two functions intersect at (x₀,f(x₀)) because our conditions mandate that f(x₀)=g(x₀). Because they have equal derivatives at that point, they must be tangent to the same line at this point. Since their second derivatives are not equal at this point, one must be accelerating faster than the other. This means that one on both sides of the point changes its slope more quickly than the other. As such, there cannot be another point in which the two functions touch until f''(x)-g''(x) has an opposite sign to f''(x₀)-g''(x₀), where the slower function can catch up to the other. Therefore, the two functions are tangent to each other.

Example:

sin(x) and x²+x are tangent to each other at x=0.

Let f(x)=sin(x) and g(x)=x²+x

f'(x)=cos(x) and g'(x)=2x+1

f''(x)=-sin(x) and g''(x)=2

f(0)=sin(0)=0 and g(0)=0²=0

f'(0)=cos(0)=1 and g'(0)=2(0)+1=1

f''(0)=-sin(0)=0 and g''(0)=2

The dictionary of obscure sorrows defines “moment of tangency” as a glimpse of what might have been. By the way, this channel is so good, and soothing definitely a recommendation.

Speaking of words. I’m so grateful that “despair” is a word. As it usually goes with me and English words, I usually hear a word a lot. I hear them in movies, or read them in books. and since I’m too lazy to look their meaning up, I usually just skip them and try to guess the general meaning of the sentence. But when a word is repeated enough, my mind catches it. And I start using the word without ever being sure of what it means. I first use consciously like my gut us telling me that this word fits here. I have no evidence whatsoever, but the word is coming from my heart so I say it and hope to God that’s enough to give it the meaning I intended :D. Despair is one of those words. I never really googled it, but I found myself using it a lot. It’s actually so cool. To me it resembles the feeling of something that’s just hanging in the air, in a vast universe without a known cause or purpose. And I’m like that a lot. I often feel as though I’m just a little thing floating over an atmosphere of mystery. Floating in emptiness, in void, in ..well.. in despair. 

 To a lot of people, being vulnerable is just a moment of weakness. Or at most a state of sadness or annoying tenderness. To me, it’s a state of existence. Being vulnerable is everything. We are vulnerable to our thoughts, we are vulnerable to our emotions. We are vulnerable to our physique, we are vulnerable to who we are. It took me a while to stop hating myself for being that vulnerable, and realize that that’s just who we are as humans. These moments of weakness are just the moments we so consciously realize this fact.