Please solve this questions below without ai and step by step.

$13\mathrm{.}2-9.$ Consider the following nonlinear programming problem:

Maximize $f(x)=x_1+x_2,$

subject to

$x_1^2+x_2^{2}1$

and

$x_{1}0,x_{2}0.$

$($a$)$ Verify that this is a convex programming problem.

$($b$)$ Solve this problem graphically.

$13\mathrm{.}2-10.$ Consider the following nonlinear programming problem:

Minimize $,Z=x_1^4+2x_2^2,$

subject to

$x_1^2+x_2^{2}2.$

$($No nonnegativity constraints.$)$

$($a$)$ Use geometric analysis to determine whether the feasible region is a convex set.

$($b$)$ Now use algebra and calculus to determine whether the feasible region is a convex set.

$13\mathrm{.}3-3.$ Consider the following nonlinear programming problem:

$]10[10$

and

$x_{1}0,x_{2}0$

$($a$)$ Of the special types of nonlinear programming problems described in Sec. $13\mathrm{.}3,$ to which type or types can this particular problem be fitted? Justify your answer.

$($b$)$ Now suppose that the problem is changed slightly by replacing the nonnegativity constraints by $x_{1}1$ and $x_{2}1.$ Convert this new problem to an equivalent problem that has just two functional constraints, two variables, and two nonnegativity constraints.