This question relates to the book Varian Microeconomic Analysis 3rd edition exercise 1.1. Much like this question but my emphasize is different.
Q: True or False? If V(y) is a convex set, then the associated production set must be convex.
Answer: as in the linked question above.
But I wonder how did Varian derive the input requirement set V(y) from his assumed production function and the associated production set? I see here there is obviously some relationship between the production function being quadratic, and the production set being the area below this graph and bounded by the x-axis, and V(y) being constrained by the inverse, the square root function, but I still do not get it. And also how to generally do it, say if we have a production function with two inputs etc.?