Problem link (1) Base case: Show that holds true. We have: . Induction step: Show that holds true leads to the fact that holds true. So the statement follows. (2) This problem is interestingly tricky. First we find the critical points of : If then (due to the fact [...]
Page 172: http://farside.ph.utexas.edu/teaching/sm1/lectures/node36.html#e4.32
To me: Page 126 of Cracking the GRE Math Subject Test. A rigorous proof can be found here (Calculus, Howard Anton). A note of stress should be on “any smooth curve C on the surface”. Below is my attempt interpretation before looking at any proofs. While it cannot be considered as a proof, it shows [...]
It took me quite a lot of time to understand the question, though. I do not post the test here as I am not sure if I am permitted to do that. This post serves for my personal use. Even though the original questions are written in Vietnamese, I will be writing my solutions in [...]
We starts with the formula (1) of the paper. We have: By chain rule, we have: We also have: – the th element of . – the th element of . So:
Show that is minimized at . Now we need to check what type of the found critical point is: So the found critical point is the global minimum.
The notation of this proof comes from the article Softmax Regression by UFLDL Tutorial. We have: When : When : Now we look carefully at the [...]