I'm confused by the notation in this question The context of this question is about the gradient descent algorithm and the question is, consider the function f : R^2 → R given by
f <- function(x) (x[1]-1)ˆ2 + 100*(x[1]ˆ2-x[2])ˆ2

Give a short mathematical proof that f has a unique minimum.
I'm unsure what the x[1] and x[2] represent here.

As others commented this is just expressing a function from using mathematical notation in R code. The function in question is $f(x) = (x_1-1)^2 + 100*(x_1^2-x_2)^2$ and as it suggests it take a vector from $\mathbb{R}^2$ and maps it to $\mathbb{R}$. Some computer languages (e.g. C or Python) start indexing the components of a vector from 0 and some from 1, as in the latter category. As such x[1] refers to the first component of vector x and so forth.
