min f(x) = $ x_1^4 + 2x_1^2x_2^2 + x_2^4 $ is an unconstrained min problem.
The first question asks to show that $(0,0)$ is the unique minimiser. I have done the following.. Would I need to add anything else to prove this?
$f'(x)=[4x_1^3 +4x_1x_2^2,4x_1^2x_2+4x_2^3]^T$
Stationary point at $f(x)= [0,0]^T$ gives us the equations
$x_1(x_1^2+x_2^2) = 0$ and $x_2(x_1^2+x_2^2)=0$
which show us that for values of $x_1$ and $x_2$ $\neq 0$ the solution to $x_1^2+x_2^2 > 0$ therefore the only stationary point is at $x_1 =x_2 = 0$.
Then stated that $f(x) >0$ for all $x\ne 0$ therefore is positive definite and a global min as $f(x)>f(0)$.
Also is quadratic convergence guaranteed when newton's method is applied with a starting point close to $(0,0)$ ?