I'm working on a problem involving multivariable calculus and Lagrange multipliers and could use some help. The task involves finding local and global extrema of a function under a given constraint.
The functions are defined as follows: $$f,g:R^3→R$$given by $$f(x)=∥x∥_2^4 +∥x∥_2$$ $$g(x)=x_1 − x_2^2$$ The goal is to find the local and global extrema of f under the constraint $$g(x)=0$$.
I know I need to set up the necessary first-order optimality conditions and find solutions of $$x^∗$$ and the corresponding Lagrange multipliers $$λ^∗$$, but I'm unsure how to proceed. Second-Order Optimality Conditions: Once I have the solutions, how do I compute the Hessian matrix of f to check the sufficient second-order optimality conditions for these solutions?
Any guidance or steps for solving this problem would be greatly appreciated.