What are the extreme points of the function $f(x_1,\dots,x_n) = x_1^1 + 2 \sum_{j=2}^n x_j^2-x_{j-1}x_j$?

Question

Consider the function $f(x_1,\dots,x_n) = x_1^1 + 2 \sum_{j=2}^n x_j^2-x_{j-1}x_j$? I'm interested in the extreme points of this map (minimum, maximum, saddle point). Approach is to calculate the gradient of $f$ and check when it vanishes. Then I would continue with the Hessian at the stationary points. The pints where the gradient vanishes are according to my calculations all points of the form $$ x_1 = x_2 = \dots = x_{n-1}, x_n = 0.5x_{n}. $$ So the gradient vanishes on a curve. The Hessian at this points seems to be positive definite. So all points on this curve are minima? Is this possible?