Given the function $$f(x_1, x_2) = (x_1 − 1)^2 + (x_2 − 1)^3$$ is $(1,1)$ a maximizer / minimizer of $f$?
The necessary conditions are fulfilled: the gradient is zero and the Hessian is PSD at $(1,1)$.
The professor teaching the course suggests me to consider the restriction to the line: $x=(1,1+2t)$. The restricted function is $f_v(t)=8t^3$. Then my professor writes that $f_v(t)$ is always increasing and then it does not have a a min/ max at $t=0$ (??)? I do not understand why this implies that $(1,1)$ is not a max nor a min.
It follows from your professor's suggestion that $f(1,1+2t)<f(1,1)$ if $t<0$ and that $f(1,1+2t)>f(1,1)$ if $t>0$. Therefore, $(1,1)$ is neither a maximum nor a minimum of $f$.